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## Geometry and Measurement

### Vector Analysis Geometry of Curves and Surfaces - Basics of Vectors

#### Q.01

'Practice (1) Find the coordinates of point Q obtained by rotating point P(-2,3) about the origin by 5/6π.'

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'(1) Let A(-3), B(7), C(2). Find the distances between points A and B, B and C, C and A, respectively. (2) Find the coordinates of the points R which divides the line segment PQ connecting two points P(-4) and Q(8) at a ratio of 1:3, the points S which divides the line segment at a ratio of 3:1 externally, and the midpoint M of segment RS.'

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'Differential Calculus 186 Conditions for Two Curves to be Tangent'

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'Illustrate the radius of the following angles and indicate in which quadrant each angle lies.'

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"I have learned about the properties of vectors in the plane and in space. Let's summarize and compare everything except for the 'reflection' of D.470."

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'Determine whether the following vectors are linearly independent or linearly dependent. (1) Vectors a=(2,1) and b=(1,-1) (2) Vectors a=(2,1) and c=(4,2)'

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'Four points A(0,1,1), B(0,2,3), C(1,3,0), D(0,1,2) are taken in space. Let the line passing through points A and B be ℓ, and the line passing through points C and D be m.'

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'(3) Line passing through a fixed point A(vector a) and perpendicular to a non-zero vector n'

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'(1) Line parallel to non-zero vector d passing through fixed point A(vector a) is represented as p=a+td, where d is the direction vector of the line'

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'Find the position vectors of the 26 intersections.'

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'Various methods of finding the position vector of an intersection point'

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'Find the range of existence of the endpoint of a vector. (4)'

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'Find the equation of a line passing through the point (1,2,-3) and parallel to the vector d=(3,-1,2).'

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'(1) Let there be 4 different points A, B, C, D on a plane and a point O not on the line AB. If OA=a, OB=b, OC=3a-2b, and OD=-3a+4b, then prove that AB is parallel to CD.'

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"In a cube with side length 1, denoted as ABCD-A'B'C'D', let points P, Q, R divide edges AB, CC', D'D' in the ratio of a:(1-a), and define vectors AB=x, AD=y, AA'=z. Given that 0<a<1, (1) express vectors PQ and PR in terms of vectors x, y, and z. (2) Find the ratio of |vector PQ| to |vector PR|. (3) Determine the angle between vectors PQ and PR."

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'In parallelogram ABCD, if 2 times vector BP equals vector BC, and 2 times vector AQ plus vector AB equals vector AC, what is the shape of quadrilateral ABPQ?'

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'In an equilateral triangle ABC with side length 2, let L, M, and N be the midpoints of the sides AB, BC, and CA respectively. Find all of the following vectors represented by the 6 points A, B, C, L, M, N:'

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'Applications of vectors\nA point P lies on the line AB if and only if \ \\overrightarrow{OP} = s\\overrightarrow{OA} + t\\overrightarrow{OB}, s + t = 1 \ where s and t are real numbers.'

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'When two linearly independent vectors \ \\overrightarrow{\\mathrm{OA}}, \\overrightarrow{\\mathrm{OB}} \ are defined on a plane, any point \ \\mathrm{P} \ can be uniquely represented as \ \\overrightarrow{\\mathrm{OP}}=s \\overrightarrow{\\mathrm{OA}}+t \\overrightarrow{\\mathrm{OB}} \\quad(s, t \ are real numbers \\( ) \\ldots \\ldots \\cdot(A) \\). In this case, the pair of real numbers \\( (s, t) \\) is called the oblique coordinates, and the point defined by (A) is denoted as \\( \\mathrm{P}(s, t) \\). In particular, when \ \\overrightarrow{\\mathrm{OA}} \\perp \\overrightarrow{\\mathrm{OB}},|\\overrightarrow{\\mathrm{OA}}|=|\\overrightarrow{\\mathrm{OB}}|=1 \, the oblique coordinates will become the \ x y \ coordinates with the extension of \ \\overrightarrow{\\mathrm{OA}} \ as the \ x \ axis and the extension of \ \\overrightarrow{\\mathrm{OB}} \ as the \ y \ axis. \\n【Basic Example 38(1)】\ \\overrightarrow{\\mathrm{OP}}=s \\overrightarrow{\\mathrm{OA}}+t \\overrightarrow{\\mathrm{OB}}, s+2 t=3 \\ldots \\ldots \ That is, the points \ \\mathrm{P} \ that satisfy \\( \\mathrm{P}(s, t), s+2 t=3 \\) lie on the line in the Cartesian coordinate plane given by \ x+2 y=3 \. The intersections of this line with the coordinate axes are \\( \\mathrm{C}(3,0) \\) and \\( \\mathrm{D}\\left(0, \\frac{3}{2}\\right) \\). Considering the points C, D with the same coordinates in the oblique coordinate plane, we have \ \\overrightarrow{\\mathrm{OA}}=\\frac{1}{3} \\overrightarrow{\\mathrm{OC}}, \\overrightarrow{\\mathrm{OB}}=\\frac{2}{3} \\overrightarrow{\\mathrm{OD}} \Thus, the condition equation (*) for point \ \\mathrm{P} \ becomes \ \\overrightarrow{\\mathrm{OP}}=\\frac{s}{3} \\overrightarrow{\\mathrm{OC}}+\\frac{2}{3} t \\overrightarrow{\\mathrm{OD}},\\frac{s}{3}+\\frac{2}{3} t=1 \, and the range for the existence of point \ \\mathrm{P} \ is the line CD.'

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'When \ \\vec{x}=2\\vec{a}-3\\vec{b}-\\vec{c}, \\vec{y}=-4\\vec{a}+5\\vec{b}-3\\vec{c} \, express \ \\vec{x}-\\vec{y} \ in terms of \ \\vec{a}, \\vec{b}, \\vec{c} \.'

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'In space, find the unit vector t that is orthogonal to the x-axis, forms a 45-degree angle with the positive z-axis, and has a positive y-component.'

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'When a point P moves on the plane and its coordinates (x, y) are functions of time t, answer the following questions:\n1. Derive the vector equation representing velocity.\n2. Derive the vector equation representing acceleration.'

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'Find the vector equation of the following lines: (a) Passes through point A(1,2,3) and is parallel to vector d=(2,3,-4). (b) Passes through points A(2,-1,1) and B(-1,3,1).'

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'Find the position vector of the intersection point of the lines.'

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'Collinearity condition\nWhen two points A, B are different\nWhen point P is on the line segment AB\n\ \\Leftrightarrow \\overrightarrow{\\mathrm{AP}}=k \\overrightarrow{\\mathrm{AB}} \ for some real number k'

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'Given three points A(6, π/3), B(4, 2π/3), C(2, -3π/4), find the following:'

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'Let the polar coordinates of point A be (r₁, θ₁), and the polar coordinates of point B be (r₂, θ₂). Find the distance AB between point A and point B.'

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'Explain the conditions for vectors to be parallel.'

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'(2) Let D, E, and F be points on the line segments OA, OB, and OC respectively, such that OD = 1/2 · OA, OE = 2/3 · OB, and OF = 1/3 · OC. If the plane containing the three points D, E, F intersects the line OQ at point R, then express vector OR in terms of vectors a, b, and c.'

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'Let the midpoints of the sides AB, BC, CD, DA of quadrilateral ABCD be K, L, M, N respectively, and the midpoints of the diagonals AC, BD be S, T respectively. (1) If the position vectors of vertices A, B, C, D are a, b, c, d respectively, express the position vector of the midpoint of segment KM using a, b, c, d. (2) By expressing the position vectors of the midpoints of segments LN, ST using a, b, c, d, prove that the three segments KM, LN, ST intersect at a single point.'

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'(1) \ \\overrightarrow{DG}=\\frac{1}{2 t} \\overrightarrow{DA}+\\frac{1}{2 t} \\overrightarrow{DB}+\\frac{t-2}{2 t} \\overrightarrow{DC} \'

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'Conditions for being collinear, coincident\n(1) Collinear conditions\nWhen two different points A, B, if a point P lies on the line AB, then there exists a real number k such that vector AP = k vector AB.'

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'In the cube OAPB-CRSQ, let 𝑝=⃗OP, 𝑞=⃗OQ, 𝑟=⃗OR. Express ⃗OA in terms of 𝑝, 𝑞, 𝑟.'

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'In the parallelepiped ABCD-EFGH, let P be the midpoint of the diagonal AG, and let vector AB be a, vector AD be b, and vector AE be c. Express the vectors AC, AG, BH, and CP in terms of a, b, and c.'

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'Find the position vectors of 26 intersection points'

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'Given \ \\mathrm{AB}=3, \\mathrm{AD}=4 \, there is a rectangle \ \\mathrm{ABCD} \. If \ \\overrightarrow{\\mathrm{AB}}=\\vec{b}, \\overrightarrow{\\mathrm{AD}}=\\vec{d} \, express the unit vector parallel to \ \\overrightarrow{\\mathrm{BD}} \ in terms of \ \\vec{b}, \\vec{d} \.'

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'Explain the fundamental concepts. In particular, provide a detailed explanation of position vectors, midpoints of line segments, and centroids of triangles.'

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'Let z be a complex number. Find the range of points for z that form an acute triangle in the complex plane with points A(1), B(z), and C(z^2), and illustrate it.'

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'(1) Explain the following vector operations regarding the basic concepts of space vectors.\n\n- Equality\n- Addition\n- Subtraction\n- Inverse vector\n- Zero vector\n- Scalar multiplication'

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'Explain the vector difference and its representation.'

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'Basic Concepts\n3. Position Vector of the Centroid of a Triangle\nLet points A(𝑎⃗), B(𝑏⃗), C(𝑐⃗) be the vertices of triangle ABC, and let G be the position vector of the centroid. Then\n𝑔⃗=1/3(𝑎⃗+𝑏⃗+𝑐⃗)'

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'(2) Line passing through two different points A(𝐚) and B(𝐛)'

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'Exercise 1 Find all the following vectors represented using the 6 vertices of the regular hexagon ABCDEF with a side length of 1 and the intersection point O of the diagonals AD and BE.'

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'For triangle OAB, let OP = sOA + tOB. Determine the range of existence of point P when the real numbers s, t satisfy the following conditions.'

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'What is the relationship between these two vectors?'

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"In triangle ABC with vertices A(a), B(b), C(c), let D be the point dividing side BC in the ratio 2:3, and E be the point dividing side BC externally in the ratio 1:2. Let G be the centroid of triangle ABC and G' be the centroid of triangle AED. Express the following vectors in terms of a, b, and c.\n(1) Position vectors of points D, E, G'\n(2) GG'"

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'Find the distance between the origin O and the point P(2,3,1).'

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'69 (2) is -2 in the x-axis direction and -3 in the y-axis direction'

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'69 (1) times in the x-axis direction by 3/2, in the y-axis direction by 5/4'

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'(1) Move 4 units along the x-axis and -7 units along the y-axis\n(2) Move -5/2 units along the x-axis and -35/4 units along the y-axis'

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"Plot the points P(z), A(α), P'(-z), B(z+α), C(z-α) on the complex plane, where z=3+2i and α=1-i."

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'For the points A(1,2,3), B(-3,2,-1), and C(-4,2,1), find the following:'

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'In rectangle ABCD, AB = 3 and AD = 4. Let AB vector be b and AC vector be c. (1) If E is the midpoint of side AD, express the vector DE using b and c. (2) Express a unit vector d in the same direction as c using c.'

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'Vector addition, subtraction, scalar multiplication'

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'Relationship between points and vectors in space\nFor two points \\( \\mathrm{A}(a_{1}, a_{2}, a_{3}), \\mathrm{B}(b_{1}, b_{2}, b_{3}) \\),\n\\[\n\egin{array}{l}\n\\overrightarrow{\\mathrm{AB}}=\\left(b_{1}-a_{1}, \\quad b_{2}-a_{2}, \\quad b_{3}-a_{3}\\right) \\\\\n|\\overrightarrow{\\mathrm{AB}}|=\\sqrt{\\left(b_{1}-a_{1}\\right)^{2}+\\left(b_{2}-a_{2}\\right)^{2}+\\left(b_{3}-a_{3}\\right)^{2}}\n\\end{array}\n\\]'

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'Chapter 2 Vector in Space If the angle between two vectors \\( \\vec{a}=(1,2,-1) \\) and \\( \\vec{b}=(-1, x, 0) \\) is \ 45^{\\circ} \, find the value of \ x \.'

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'(1) Find the equation of a line passing through point A(3,1) and perpendicular to the vector n=(3,-7).'

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'(2) \ 4 \\overrightarrow{\\mathrm{AQ}}+\\overrightarrow{\\mathrm{BQ}}+2 \\overrightarrow{\\mathrm{CQ}}=\\overrightarrow{0} \'

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'For the vectors $\\ vec a, \\ vec b$ on the right, depict the following vectors:'

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'In the regular hexagon ABCDEF, with AB→=a and AF→=b. Represent the following vectors in terms of a and b. (1) CE→ (2) EA→ (3) AD→'

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'Find the equation of the line passing through point C(1,-5) and perpendicular to the line AB, where A(3,1) and B(-2,2), using vectors.'

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'7 \\overrightarrow{\\mathrm{OC}}=\\frac{4}{9} \\overrightarrow{\\mathrm{OA}}+\\frac{1}{6} \\overrightarrow{\\mathrm{OB}}'

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"Let's focus on the results of the example on the previous page."

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'Explain the component representation of vectors in space.'

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'Find the equation of a line that satisfies the following conditions using vectors:\n1. Passes through point A(-3,5) and is parallel to vector d=(1,-sqrt(3))\n2. Passes through points A(-7,-4) and B(5,5)'

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'In tetrahedron OABC, let OA=a, OB=b, and OC=c. Let M be the midpoint of AB, N be the point that divides BC in the ratio 3:1, and G be the centroid of triangle OAB. Express vectors MN and GN in terms of a, b, and c.'

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"Let's consider a straight line passing through a point and having a given slope (direction). Let g be the line passing through point A(\\\vec{a}\) and parallel to a non-zero vector \\\vec{d}\. For any point P(\\\vec{p}\) on the line g (excluding point A), the following holds true."

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'Math problem: Finding the position vector of the centroid G of the triangle OAB. Since point G is the centroid of the triangle, the position vector of point G can be calculated as follows.'

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'For the points \\( \\mathrm{A}(1,2,3), \\mathrm{B}(-3,2,-1), \\mathrm{C}(-4,2,1) \\), find the following:\n(1) Distance between the points \ \\mathrm{B}, \\mathrm{C} \\n(2) Coordinates of the point \ \\mathrm{P} \ that divides the segment \ \\mathrm{BC} \ in the ratio 1:3\n(3) Coordinates of the point \ \\mathrm{Q} \ that divides the segment \ \\mathrm{AB} \ externally in the ratio 2:3\n(4) Coordinates of the midpoint R of the segment CA\n(5) Coordinates of the centroid G of the triangle \ \\triangle \\mathrm{PQR} \'

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'For points A(0,3,7), B(3,-3,1), C(-6,2,-1), find the following:\n(1) Distance between points A and B\n(2) Coordinates of a point that divides the line segment AB in the ratio 2:1\n(3) Coordinates of a point that divides the line segment AB externally in the ratio 3:2\n(4) Coordinates of the midpoint of the line segment BC\n(5) Coordinates of the centroid of triangle ABC'

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"The line perpendicular to vector \ \\vec{n} \\nFinally, let's consider expressing the line using dot product.\nPassing through point \\( \\mathrm{A}(\\vec{a}) \\), and a non-zero vector \ \\overrightarrow{0} \ perpendicular to vector \ \\vec{n} \ is denoted as line \ g \, and any point on line \ g \ is denoted as \\( \\mathrm{P}(\\vec{p}) \\) such that \ \\vec{n} \\perp \\overrightarrow{\\mathrm{AP}} \ or \ \\overrightarrow{\\mathrm{AP}}=\\overrightarrow{0} \\n\\[\n\egin{array}{l}\n\\Longleftrightarrow \\vec{n} \\cdot \\overrightarrow{\\mathrm{AP}}=0 \\\\\n\\Longleftrightarrow \\vec{n} \\cdot(\\vec{p}-\\vec{a})=0\n\\end{array}\n\\]\n(D) represents the vector equation of a line passing through point \ \\mathrm{A} \ and perpendicular to vector \ \\vec{n} \. Furthermore, \ \\vec{n} \ is referred to as the normal vector of line \ g \.\n\ -\\overrightarrow{\\mathrm{AP}}=\\overrightarrow{0} \ holds true only when point P coincides with point A.\nThe normal vector of line \ g \ is perpendicular to it.\nNow, let's solve a vector equation problem."

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'Find the position vectors of the internal and external division points.'

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'The pyramid \\\mathrm{OABCD}\ with base \ \\mathrm{ABCD} \ satisfies \ \\overrightarrow{\\mathrm{OA}}+\\overrightarrow{\\mathrm{OC}}=\\overrightarrow{\\mathrm{OB}}+\\overrightarrow{\\mathrm{OD}} \, and for four real numbers \ p, q, r, s \ different from 0, define the four points \ \\mathrm{P}, \\mathrm{Q}, \\mathrm{R}, \\mathrm{S} \ as \ \\overrightarrow{\\mathrm{OP}}=p \\overrightarrow{\\mathrm{OA}} \, \ \\overrightarrow{\\mathrm{OQ}}=q \\overrightarrow{\\mathrm{OB}}, \\overrightarrow{\\mathrm{OR}}=r \\overrightarrow{\\mathrm{OC}}, \\overrightarrow{\\mathrm{OS}}=s \\overrightarrow{\\mathrm{OD}} \. Show that if \ \\mathrm{P}, \\mathrm{Q}, \\mathrm{R}, \\mathrm{S} \ are coplanar, then \ \\frac{1}{p}+\\frac{1}{r}=\\frac{1}{q}+\\frac{1}{s} \.'

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'3 (1) \ \\frac{2 \\vec{a}+3 \\vec{b}}{5} \\n(2) \ 2 \\vec{a}-\\vec{b} \\n(3) \ \\frac{\\vec{a}+\\vec{b}}{2} \'

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'Given points P(5,-3,7) and Q(7,1,2), find the components and magnitude of vector PQ.'

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'Plot the position of the following points in polar coordinates:'

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'Describe the range of existence of point P as it moves satisfying s + t ≤ 1, s ≥ 0, t ≥ 0 conditions.'

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'Find the necessary and sufficient condition for a point P (vector 𝑝) to lie on the straight line AB passing through two distinct points A (vector 𝑎) and B (vector 𝑏).'

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'Important Example 63 | Length of Common Perpendicular\nIn the coordinate space, let point A(1,3,0) lie on a line l parallel to vector a=(-1,1,-1), and point B(-1,3,2) lie on a line m parallel to vector b=(-1,2,0). Let P be a point on line l and Q be a point on line m. Find the minimum value of the magnitude |PQ| of vector PQ, and the coordinates of points P and Q at that instance.'

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'The condition for vectors to be parallel (\ \\vec{a} \\neq \\overrightarrow{0}, \\vec{b} \\neq \\overrightarrow{0} \) is that there exists a real number \ k \ such that \ \\vec{a} / / \\vec{b} \\Leftrightarrow \\vec{b}=k \\vec{a} \'

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'For △OAB, let →OP=→OA+t→OB. Find the range of existence of point P when real numbers s, t satisfy the following relations: (1) 3s+t=2 (2) 2s+t≤1, s≥0, t≥0'

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'Given line segment AB and point P, when AP + 3BP + 4AB = 0, where is point P located?'

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'Practice 38:\n(1) (A) \\overrightarrow{\\mathrm{AB}}=(-2,1,2), \\overrightarrow{\\mathrm{AC}}=(a-1,-2,3) therefore\n\\overrightarrow{\\mathrm{AB}}\\cdot \\overrightarrow{\\mathrm{AC}}=-2\\cdot (a-1)+1\\cdot(-2)+2\\cdot 3=-2a+6\n\\left|\\overrightarrow{\\mathrm{AB}}\\right|=\\sqrt{(-2)^{2}+1^{2}+2^{2}}=3\n\\left|\\overrightarrow{\\mathrm{AC}}\\right|=\\sqrt{(a-1)^{2}+(-2)^{2}+3^{2}}=\\sqrt{a^{2}-2a+14}'

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"Let's consider the midpoint of the diagonal RT as G, and take vector OP=p, vector OR=r, and vector OS=s."

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'Explain the rules of vector operations and use those rules to prove properties.'

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'Find the position vector p of point P, which divides the line segment connecting points A(a) and B(b) into m:n.'

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'In this case, with O as the origin, $\\overrightarrow{OH} = \\overrightarrow{OP} + \\overrightarrow{PH} = (4,4,2) + (-1,-2,-4) = (3,2,-2)$ therefore $H(3,2,-2)$'

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'Explain and indicate the conditions for vectors to be parallel.'

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'Find the position vector g of the centroid G of triangle ABC, with A(a), B(b), C(c) as the vertices.'

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'If \ \\vec{a} \\neq \\overrightarrow{0} \, \ \\vec{b} \\neq \\overrightarrow{0} \, and \ \\vec{a} \\times \\vec{b} \, then any vector \ \\vec{p} \ can be uniquely represented as \ \\vec{p}=s \\vec{a}+t \\vec{b} \, where \ s, t \ are real numbers.'

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'Page 15 | Equation of Vectors and Position of Points (2)'

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'Position vector and collinearity condition\nThe position vector of a point that divides the line segment \ \\mathrm{AB} \ with points \\( \\mathrm{A}(\\vec{a}) \\) and \\( \\mathrm{B}(\\vec{b}) \\) in the ratios of \ m: n \ is as follows: internal division: \ \\cdots \\cdots \\frac{n \\vec{a} + m \\vec{b}}{m + n} \, external division: \ \\cdots \\cdots \\frac{-n \\vec{a} + m \\vec{b}}{m - n} \\nCollinearity condition\nWhen the points \ \\mathrm{A}, \\mathrm{B} \ are different, a real number \ k \ exists such that the point \ \\mathrm{P} \ lies on the line \ \\mathrm{AB} \\n\ \\Leftrightarrow \\overrightarrow{\\mathrm{AP}} = k \\overrightarrow{\\mathrm{AB}} \ for some real number \ k \'

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'Based on the information given, find the expressions and relationships of various vectors.'

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'Explain how an arbitrary vector p can be decomposed using two non-parallel vectors a and b.'

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'Find the parametric equations for the following curve.'

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'A directed line segment AB is represented as vector →AB. Additionally, vectors can also be represented using a single letter with an arrow, such as →a, →b. How do we represent the magnitude of vectors →AB, →a?'

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'Given the coordinates of points and the components of a vector \\( \\mathrm{A}(a_1, a_2), \\mathrm{B}(b_1, b_2) \\)'

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'For the line segment connecting point A(vector a) and point B(vector b) as AB, express the position vectors of the following points in terms of vectors a and b.'

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'Let the coordinates of point C be \\((x, y, z)\\). Using the condition that the quadrilateral \ \\mathrm{ABCD} \ is a parallelogram, find the coordinates of C.'

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'Find the range of movement of point P under the following conditions.'

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'Please prove that the magnitude of the real multiple k*𝐚 is k times the magnitude of 𝐚, and the direction is the same as 𝐚 when k > 0 and opposite when k < 0.'

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'Find the vector equation of a line passing through point A(𝑎→) and parallel to vector 𝑑(𝑑 ≠ 𝑎→).'

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'In triangle OAB, find the range of points P that satisfy the following conditions.'

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'Given line segment AB and point P. When the following equation holds true, where is point P located?'

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'Translate the given text into multiple languages.'

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'In the XY plane, the coordinates of point A are (1,0), the coordinates of point B are (cos 𝛼, sin 𝛼) (0 ≤ 𝛼 < 2π), and the coordinates of point C are (cos 𝛽, sin 𝛽) (0 ≤ 𝛽 < 2π) without loss of generality. Therefore, OA=(1,0), OB=(cos 𝛼, sin 𝛼), OC=(cos 𝛽, sin 𝛽).'

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'Let p be a positive constant, and let vector a=(1,1) and b=(1,-p). Now, if the angle between vectors a and b is 60 degrees, find the value of p.'

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'Find the components of each vector, where 21 \\\overrightarrow{AC}=\\vec{a}+\\vec{b}\, \\\overrightarrow{AG}=\\vec{a}+\\vec{b}+\\vec{c}\, \\\overrightarrow{BH}=-\\vec{a}+\\vec{b}+\\vec{c}\, and \\\overrightarrow{CP}=-\\frac{1}{2}\\vec{a}-\\frac{1}{2}\\vec{b}+\\frac{1}{2}\\vec{c}\.'

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'Example 37 | Unit vectors perpendicular to two vectors'

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'In acute-angled triangle ABC, with A(→a), B(→b), C(→c), BC=a, CA=b, AB=c. If the incenter of angle A is denoted as IA(→iA), express the vector →iA in terms of →a, →b, and →c.'

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'(1) Find the distance between point P(0,1,4) and point Q(-4,5,0).'

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'Derive the vector equation of a straight line passing through point A (a vector) and perpendicular to vector n (n is not equal to zero vector).'

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'In triangle ABC with vertices A(a), B(b), and C(c), let point P divide side AB in the ratio 2:1, point Q divide side BC externally in the ratio 3:2, and point R divide side CA externally in the ratio 1:3. Let G be the centroid of triangle PQR. Express the following vectors in terms of a, b, and c: (1) Position vectors of points P, Q, R (2) Vector PQ (3) Position vector of point G'

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'Point Q divides the line segment OP externally in the ratio 4:1, so'

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'Prove that the inequality |vector AP| + |vector AQ| + |vector AR| ≥ 3/√2 holds for the 4 points P(x, y), Q(y, z), R(z, x), A(0,1)(x, y, z) in real numbers.'

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'Verify the coordinates of point P (x, y) and take point Q.'

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'27 (1) \ \\overrightarrow{\\mathrm{OF}} = \\frac{3}{8} \\vec{a} \\n(2) \ \\overrightarrow{\\mathrm{OE}} = \\frac{5}{6} \\vec{a} - \\frac{2}{3} \\vec{b} \'

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'Given vectors \\(\\vec{e}_1 = (1,0)\\), \\(\\vec{e}_2 = (0,1)\\), \\\vec{a} = \\overrightarrow{OA}\, and \\\vec{b} = \\overrightarrow{OB}\ (where O is the origin), with \\\vec{a} = -3\\overrightarrow{e_1} + 2\\overrightarrow{e_2}\ and \\\vec{b} = 3\\overrightarrow{e_1} + 4\\overrightarrow{e_2}\, plot the vectors \\\vec{a}\ and \\\vec{b}\ on the coordinate plane.'

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'(1) Express \ \\overrightarrow{G U} \ as \ \\vec{p}, \\vec{r}, \\vec{s} \.'

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'Translate the given text into multiple languages.'

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'Consider an equilateral triangle ABC with side length 1 on the plane. For a point P, let vector v(P) be defined as v(P)=→PA−3→PB+2→PC. Prove: (1) v(P) is a constant vector independent of P. (2) For which figure does point P lie when |→PA+→PB+→PC|=|v(P)|.'

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'Perform vector addition, subtraction, scalar multiplication, and vector operations between two points using vector components.'

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'Let the centers of circles C1 and C2 be O1 and O2 respectively. What is the vector from center O1 to center O2?'

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'For any point P on the line, let OP = p. In this case, what is the vector equation of the line?'

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'Translate the given expression into multiple languages.'

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'When four points O, A, B, C are not in the same plane, if $\\overrightarrow{OA}=\\vec{a}$, $\\overrightarrow{OB}=\\vec{b}$, $\\overrightarrow{OC}=\\vec{c}$, then any vector $\\vec{p}$ can be uniquely represented as $\\vec{p}=s\\vec{a}+t\\vec{b}+u\\vec{c}$, where $s$, $t$, $u$ are real numbers.'

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'Find the coordinates of point Q after rotating point P(3, -1) around point A(-1, 2) as the center by -π/3.'

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'Please provide the formula to calculate the distance AB between points A(r1, θ1) and B(r2, θ2).'

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'Plot the following points in polar coordinates and find the Cartesian coordinates.'

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'(1) Find the coordinates of point B after rotating point A(2,1) around the origin O by π/4 radians.\n(2) Point P was the center of rotation when point A(2,1) was rotated by π/4 radians to the coordinates (1-√2, -2+2√2). Find the coordinates of point P.'

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'Point P moves along the circumference of a circle with radius r centered at the origin O, starting from fixed point P0, such that OP rotates at a constant angular velocity ω per second.'

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'Understanding of basics of vectors (definition of vectors, properties, and basic operations).'

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'In the coordinate plane, there are three fixed points A, B, C, and a moving point P, with vector AB=(3,1), vector BC=(1,2), and vector AP represented as (2t, 3t) using the real number t.'

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'Important Topic 40 | Comparison of Vector Magnitudes'

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'Find the distance between the following two points.'

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'In \ \\triangle OAB \, find the range of the existence of point \ P \ that satisfies the following equations. \\n(1) \ \\overrightarrow{OP}=s\\overrightarrow{OA}+t\\overrightarrow{OB}, 3s+4t=4 \\\n(2) \ \\overrightarrow{OP}=s\\overrightarrow{OA}+3t\\overrightarrow{OB}, 0\\leqq 2s+5t\\leqq 1, s\\geqq 0, t\\geqq 0 \'

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'Show that the following equations hold for the tetrahedron ABCD: (1) $\\overrightarrow{AB}+\\overrightarrow{BD}+\\overrightarrow{DC}+\\overrightarrow{CA}=\\overrightarrow{0}$ (2) $\\overrightarrow{BC}-\\overrightarrow{DA}=\\overrightarrow{AC}-\\overrightarrow{DB}$'

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'Find the equation of the line passing through two different points A(\ \\vec{a} \) and B(\ \\vec{b} \).'

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'Prove the following equation: (1) 2 \ \\overrightarrow{\\mathrm{AB}}+\\overrightarrow{\\mathrm{DC}}+\\overrightarrow{\\mathrm{EF}}=\\overrightarrow{\\mathrm{DB}}+\\overrightarrow{\\mathrm{EC}}+\\overrightarrow{\\mathrm{AF}} \'

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'About the solutions for position vectors and conditions for being on the same plane'

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'Chapter 1 Vectors on a Plane - In triangle OAB, determine the range of points P that satisfy the following equations:\n1) OP = sOA + tOB, s + t = 1/3, s ≥ 0, t ≥ 0\n2) OP = sOA + tOB, 3s + 2t = 4, s ≥ 0, t ≥ 0'

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'In the PR coordinate space, there are 4 points O(0,0,0), A(3,-2,-1), B(1,1,1), C(-1,4,2). Find the vector p that is perpendicular to both vectors OA and BC, with a magnitude of 3√3.'

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'Find the range of existence of point P that satisfies the following conditions.'

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'The range of the existence of points that satisfy the vector equation and the intersection coordinates'

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'Given four points A(1,1,-2), B(-2,1,2), D(3,-1,-3), E(9, a, b).'

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'Prove that the following equations hold in tetrahedron ABCD: (1) $\\overrightarrow{\\mathrm{AB}}+\\overrightarrow{\\mathrm{BD}}+\\overrightarrow{\\mathrm{DC}}+\\overrightarrow{\\mathrm{CA}}=\\overrightarrow{\\mathrm{0}}$ (2) $\\overrightarrow{\\mathrm{BC}}-\\overrightarrow{\\mathrm{DA}}=\\overrightarrow{\\mathrm{AC}}-\\overrightarrow{\\mathrm{DB}}$'

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'Example 21 | Representation of space vectors\nIn the parallelepiped ABCD-EFGH, let the midpoint of the diagonal AG be P, and let →AB=𝑎, →AD=𝑏, →AE=𝑐. Express →AC, →AG, →BH, →CP in terms of 𝑎, 𝑏, 𝑐.'

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'In tetrahedron OABC, let vector a=OA, vector b=OB, and vector c=OC. Let L, M, N, P, Q, and R be the midpoints of segments OA, OB, OC, BC, CA, and AB respectively. Let vector p=LP, vector q=MQ, and vector r=NR.'

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'Explain the condition for vectors to be parallel.'

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'Use the following equations to demonstrate properties of vectors:'

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'In triangle ABC with vertices A(a), B(b), and C(c), where point P divides side AB internally in the ratio 2:1, point Q divides side BC externally in the ratio 3:2, and point R divides side CA externally in the ratio 1:3, and G is the centroid of triangle PQR. Express the following vectors in terms of a, b, and c: (1) Position vectors of points P, Q, and R (2) Vector PQ (3) Position vector of point G'

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'Consider a circle with center O. There are 3 points A, B, and C on the circumference of this circle such that OA vector + OB vector + OC vector = 0. Prove that triangle ABC is an equilateral triangle.'

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'Vector decomposition In parallelogram ABCD, point E divides side BC internally in the ratio 2:1, point F is the intersection of the diagonals AC and BD, and point G is the intersection of segments AE and BD. Let vector AB=b and vector AD=d. (1) Express vectors AE, AF, GC in terms of b and d. (2) If vector AE=e and vector AF=f, then express vector BD in terms of e and f.'

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'Linear independence and linear dependence of vectors in space'

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'When the three vectors a, b, and c are not collinear, find the equation of the plane passing through these three points.'

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'In \ \\triangle \\mathrm{OAB} \, find the range of points \ \\mathrm{P} \ that satisfy the following equations:'

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'For the vectors a, b, c, plot the following vectors:\n1. a + b\n2. a - c\n3. 3b\n4. -2c'

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'Let \\( \\vec{a}=(1,-1,2) \\) and \\( \\vec{b}=(1,1,-1) \\). Find the minimum magnitude of \ \\vec{a}+t \\vec{b} \ (where \ t \ is a real number) and the corresponding value of \ t \.'

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'Applications of vectors Let s, t, u be real numbers with the same plane. A point P (p) is on the plane determined by three points A (a), B (b), and C (c) if and only if ⇔CP⃗ = sCA⃗ + tCB⃗ ⇔ p⃗ = sa⃗ + tb⃗ + uc⃗, s + t + u = 1'

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'Example 38 Problem on angles formed by vectors\n(1) In space, there are fixed points A(0,4,2) and B(2√3, 2,2), and a moving point P(0,0,p). Find the maximum value of ∠APB denoted as θ(0° ≤ θ ≤ 180°) and the corresponding value of p.\n(2) For vectors a=(3,-4,12), b=(-3,0,4), and c=a+tb, find the real value of t when the angle between c and a is equal to the angle between c and b.'

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'Important Example 61: Equations of Lines\nFind the equations of the following lines:\n(1) Passing through point A(1,3,-2) and parallel to vector d=(3,2,-4)\n(2) Passing through points A(0,1,1) and B(-1,3,1)\n(3) Passing through point A(-3,5,2) and parallel to vector d=(0,0,1)'

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'Any point on the line is denoted as P(x, y) with t as a parameter.'

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'Find the equation of the line \ \\ell \ passing through point A(\ \\vec{a} \) and parallel to the non-zero vector \ \\vec{d} \.'

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'Any vector \ \\vec{p} \ on the plane can be expressed in terms of two vectors \\( \\vec{a}, \\vec{b} (\\vec{a} \\neq \\overrightarrow{0}, \\vec{b} \\neq \\overrightarrow{0}, \\vec{a} \\times \\vec{b}) \\), as follows:'

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'Perform the following vector operations: 1. Find the sum of vector A = (3, 4) and vector B = (1, 2). 2. Find the difference between vector A = (3, 4) and vector B = (1, 2).'

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'The vector equation of a plane defined by three non-collinear points A(⃗a), B(⃗b), C(⃗c) and an arbitrary point P(⃗p), with s, t, u as real numbers, is given by ⃗p=s⃗a+t⃗b+u⃗c, s+t+u=1 or ⃗p=s⃗a+t⃗b+(1-s-t)⃗c'

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'When two vectors a and b have the same direction and magnitude, these two vectors are considered equal.'

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'Find the value of the real number $t$ for which the angle between the vectors $\\vec{a} + t \\vec{b}$ and $\\vec{b} + t \\vec{a}$ is $120^{\\circ}$, given that $\\vec{a} = (3, 4, 5)$ and $\\vec{b} = (7, 1, 0)$.'

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'Find the complex numbers representing the following points. (1) The midpoint of the line segment AB connecting the points A(-3+6i) and B(5-8i) (2) The point P which divides the line segment AB connecting the points A(2-3i) and B(-7+3i) in the ratio 2:1, and the point Q which divides externally.'

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'Find the angle θ formed by the two planes. Where 0° ≤ θ ≤ 90°. (1) 4x-3y+z=2, x+3y+5z=0 (2) x+y=1, x+z=1 (3) -2x+y+2z=3, x-y=5'

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'Let point P divide side OA of acute-angled triangle OAB in the ratio k:(1-k), and point Q divide side OB in the ratio l:(1-l). Also, let R be the intersection of AQ and BP. Let OA be represented by vector a, and OB be represented by vector b. 1) Express OP vector and OQ vector in terms of vectors a and b. 2) Express OR vector in terms of vectors a and b.'

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'For the line segment connecting points A(a) and B(b) as AB, express the position vectors of the following points in terms of a and b.'

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'Position vector and internal division points・external division points\nLet the position vector be \ \\vec{p} \ and denote the point as \\( \\mathrm{P}(\\vec{p}) \\).\nIn space, just like in a plane, the following holds:\n\nProblem 1: For points \\( \\mathrm{A}(\\vec{a}), \\mathrm{B}(\\vec{b}), \\mathrm{C}(\\vec{c}) \\), derive the following equations.\n1. \ \\overrightarrow{\\mathrm{AB}} = \\vec{b} - \\vec{a} \\n2. Find the position vector of a point dividing the line segment \ \\mathrm{AB} \ in the ratio of \ m: n \.\n3. Find the position vector of the midpoint of the line segment \ \\mathrm{AB} \.\n4. Find the position vector of the centroid G of \ \\triangle \\mathrm{ABC} \.'

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'Question 71\n(1) Find the point (-3, -1, 7).\n(2) Find the point (18/11, 26/11, 12/11).'

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'Find the range of existence of point P in triangle OAB that satisfies the following equations:\n(1) $\\overrightarrow{OP}=s \\overrightarrow{OA}+t \\overrightarrow{OB}, 1 \\leqq s+2 t \\leqq 2, s \\geqq 0, t \\geqq 0$\n(2) $\\overrightarrow{OP}=s \\overrightarrow{OA}+(s-t) \\overrightarrow{OB}, 0 \\leqq s \\leqq 1, 0 \\leqq t \\leqq 1$'

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'Find the distance between point A(r1, θ1) and point B(r2, θ2).'

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'Position Vectors, Vectors, and Shapes: Explain how position vectors and vectors form shapes.'

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'Practice (1) For two non-zero vectors \ \\vec{a} \ and \ \\vec{b} \, when \ \\vec{a}+2 \\vec{b} \ is perpendicular to \ \\vec{a}-2 \\vec{b} \, and when \ 7|\\vec{a}+2 \\vec{b}|=2|\\vec{b}| \ is true, find the angle \ \\theta \ formed by \ \\vec{a} \ and \ \\vec{b} \.'

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'Let \ \\vec{a} \ and \ \\vec{b} \ be two non-zero vectors that are orthogonal. Let the angle between \ \\vec{a}+\\vec{b} \ and \ \\vec{a}+3 \\vec{b} \ be \ \\theta \ \ 0 \\leqq \\theta \\leqq \\pi \. (1) Express \ \\sin ^{2} \\theta \ in terms of \ x \ and \ y \ where \ |\\vec{a}|=x,|\\vec{b}|=y \. (2) Find the maximum value of \ \\theta \.'

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"A vector is a quantity with both magnitude and direction. This chapter discusses vectors based on directed line segments on a plane, representing vectors with pairs of numbers (components), operations like 'dot product,' and applying vectors to geometry. Understanding the concept of 'linear independence' in vectors is crucial for preparing for future studies in mathematics, physics, economics, and other fields."

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'Plot the positions of the following points represented in polar coordinates: \\(A\\left(3, \\frac{\\pi}{6}\\right)\\), \\(B\\left(2, \\frac{3}{4} \\pi\\right)\\), \\(C\\left(1,-\\frac{2}{3} \\pi\\right)\\).'

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'Let \ 45^{\\circ} \\mathrm{O} \ be the origin, \\( \\mathrm{A}(2,1), \\mathrm{B}(1,2), \\overrightarrow{\\mathrm{OP}}=s \\overrightarrow{\\mathrm{OA}}+t \\overrightarrow{\\mathrm{OB}}(s, t \\) (where s and t are real numbers).'

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'In triangle OAB, let C be the midpoint of OA and D be the point that divides OB in a 1:3 ratio externally. If OA vector is a and 35 times the OB vector is b, find the vector equation of the following line.'

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'In parallelogram \ABCD-EFGH\, when \\\overrightarrow{AC}=\\vec{p}, \\overrightarrow{AF}=\\vec{q}, \\overrightarrow{AH}=\\vec{r}\, express \\\overrightarrow{AB}, \\overrightarrow{AD}, \\overrightarrow{AE}, \\overrightarrow{AG}\ in terms of \\\vec{p}, \\vec{q}, \\vec{r}\.'

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'Find the parametric representation of the following line with parameter t.'

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'Find the parametric representation of the following line with parameter t: passes through point B(-4,3) and parallel to vector d=(5,6).'

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'Vector equation of a circle: Vector equation of a circle with center C(c) and radius r is |p-c|=r'

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'Conditions of coordinates of points on a straight line\nProblem 6: In space, 3 points \\( \\mathrm{A}(3,2,6), \\mathrm{B}(5,-1,4), \\mathrm{C}(x, y, 0) \\) are collinear when \ x= \ \ \\square, y= \\square \ .'

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'Find the distance between the following two points.'

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'Find the components and magnitude of the vector PQ for points P(5,-3,7) and Q(7,1,2).'

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'In this example, given vectors \\( \\vec{a}=(1,3,2), \\vec{b}=(0,1,-1), \\vec{c}=(5,1,3) \\), express the vector \\( \\vec{d}=(7,6,8) \\) in the form \ s \\vec{a}+t \\vec{b}+u \\vec{c} (s, t, u \ real numbers.'

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'Derivation of the formula for the area of a triangle using vectors'

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'Illustrate the addition, subtraction, and scalar multiplication of vectors.'

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'Learn about vector equations and point positions (1).'

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'Example 14 | Equality of vectors and position of points (1)'

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'Let N be the point that divides AB in the ratio of 2:3. Let P be the intersection point of the line segment LM and ON. If a is the vector OA and b is the vector OB, express ON and OP in terms of vectors a and b.'

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'Translate the given text into multiple languages.'

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'How does the thinking differ between Approach 1 and Approach 2 for Basic Example 24?'

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'Consider a circle C with radius r and center position vector →OA on the coordinate plane with EXO as the origin. Let the position vector of a point P on the circumference be →OP. Also, consider a point B outside circle C with position vector →OB. Furthermore, let Q be the midpoint of points B and P, with position vector →OQ. Define D as the shape traced by point Q as point P moves along the circumference.\n(1) Find the vector equation representing circle C.'

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'Determine the range of existence for points satisfying the vector equation. Please answer using an example of a spatial geometric shape.'

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'Learn about vector equations and point positions (2).'

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'Find the equation of the plane passing through the point (-1,2,3) and perpendicular to the line \\frac{x-2}{4}=\\frac{y+1}{-3}=z-3.'

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'Prove that the point P lies on the line AB, where AB is determined by two different points A(a) and B(b).'

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'Given points A(2,1,0), B(1,0,1), C(0,1,2), D(1,3,7). Let E be the symmetric point of D with respect to the plane passing through points A, B, and C. Find the coordinates of point E.'

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'(1) Find the coordinates of the point symmetric to point P(-3, 4, 1) with respect to point A(1, -2, 3).'

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'Collinearity condition: When two points A, B are different, point P lies on the line AB if and only if the vector AP is equal to k times the vector AB, where k is a real number.'

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'Find the equation of the straight line passing through point A and perpendicular to vector n. (2) A(1,3), n=(-1,2)'

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'Find the parametric equations of the following lines, with parameter t.'

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'Given two points A(vector a) and B(vector b), find the position vector of the point where the line segment AB is divided into m:n.'

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'In coordinate space, shapes and vector equations: Explain the representation of shapes in coordinate space and the use of vector equations.'

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'Point P lies on the line AB if and only if there exist real numbers s and t such that the vector OP equals s times the vector OA plus t times the vector OB, and s + t = 1'

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'Find the parametric representation of the following straight lines with the parameter t. Also, express it without the parameter t.'

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'Describe the components of vectors in space, dot product: Explain the calculation of components of vectors in space and their dot product.'

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'Question 46\n(1) Find the coordinates of the point (0, 1/4, 0).\n(2) Find the coordinates of the point (0, -21, 17/2).'

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'Let a = (0,1,2), b = (2,4,6). For real numbers t, where -1 ≤ t ≤ 1, find the values of x that maximize and minimize the magnitude of x when x = a + tb.'

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'If $|\\vec{a}|=2,|\\vec{b}|=3,|2\\vec{a}-\\vec{b}|=\\sqrt{13}$, find the range of real numbers for $k$ such that $|k\\vec{a}+t\\vec{b}|>\\sqrt{3}$ holds for all real numbers $t$.'

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'In space, there is a triangle ABC with vertices A(5,0,1), B(4,2,0), C(0,1,5). (1) Find the lengths of segments AB, BC, and CA. (2) Find the area S of triangle ABC.'

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'Consider a plane α determined by three non-collinear points A (vector a), B (vector b), and C (vector c) that are not on the same line. When point P (vector p) lies on the plane α, the following vector equation holds true. Prove this.'

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'Point C divides the side \\\mathrm{OA}\ of triangle \\\triangle OAB\ in the ratio 3:1, and point D divides the side \\\mathrm{OB}\ in the ratio 4:1. Let P be the intersection of the segments \\\mathrm{AD}\ and \\\mathrm{BC}\, and Q be the intersection of the segments \\\mathrm{OP}\ and \\\mathrm{AB}\. Given that \\\overrightarrow{\\mathrm{OA}}=\\vec{a}\ and \\\overrightarrow{\\mathrm{OB}}=\\vec{b}\, express \\\overrightarrow{\\mathrm{OP}}\ in terms of \\\vec{a}\ and \\\vec{b}\ and find the ratio of BP to CP. Also, express \\\overrightarrow{\\mathrm{OQ}}\ in terms of \\\vec{a}\ and \\\vec{b}\ and determine the ratio of \\\mathrm{OP}\ to \\\mathrm{PQ}\.'

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'What is the curve represented by the polar equation r = sin αθ called? Also, show the change in the number of petals based on the value of a.'

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'In the line segment AB, when the direction is specified from point A to point B, it is called directed line segment AB. In the directed line segment AB, A is called its initial point and B is called its terminal point. The length of the line segment AB is called the magnitude or length of the directed line segment AB. Ignoring the difference in position, focusing only on the direction and magnitude is called a vector. Write down the vector represented by the directed line segment AB.'

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'(1) Find the distance between two points A(1,-1,3) and B(-1,0,1).'

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The condition for point P(\vec{p}) to be on the plane defined by three points A(\vec{a}), B(\vec{b}), and C(\vec{c}) is
$\overrightarrow{\mathrm{CP}}=s \overrightarrow{\mathrm{CA}}+t \overrightarrow{\mathrm{CB}}\ \ Longleftrightarrow \vec{p}=s \vec{a}+t \vec{b}+u \vec{c}, \ s+t+u=1$

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■ The position vector of the centroid of triangle $\mathrm{ABC}$, the position vector of the centroid G of triangle ABC, is established as follows. The position vector of the centroid of Triangle ABC, with vertices \( \mathrm{A}(\vec{a}), \mathrm{B}(\vec{b}), \mathrm{C}(\vec{c}) \), is
$\frac{\vec{a}+\vec{b}+\vec{c}}{3}$

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On the equality of vector coefficients: If ec{a}
eq \overrightarrow{0}, ec{b}
eq \overrightarrow{0}, then s ec{a} + t ec{b} = s^{\prime} ec{a} + t^{\prime} ec{b} ⇔ $s = s^{\prime}, t = t^{\prime}$

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Let lpha=x-2 i and eta=3-6 i . When the two points \( \mathrm{A}(lpha) \) and \( \mathrm{B}(eta) \) lie on the same straight line with the origin $\mathrm{O}$, find the value of the real number $x$.

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Given the triangle $riangle \mathrm{ABC}$ with vertices \( \mathrm{A}(1,1,0), \mathrm{B}(0,2,2), \mathrm{C}(1,2,1) \), find the magnitude $heta$ of the angle ngle \mathrm{BAC} .

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Parallel Vectors
Two non-zero vectors ec{a}, \ec{b} are said to be parallel if they have the same or opposite direction, and it is written as $\vec{a}/\vec{b}$. From the definition of real multiples of vectors, the following holds true.
Next, demonstrate that sample vectors ecа= лучший ), \вейвек same and equal to \( 2 \vec {b}。

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Find the angle $\theta$ between vectors $\vec{a}$ and $\vec{b}$ in the following cases.
(1) When $|\vec{a}|=1, |\vec{b}|=2 \sqrt{2}, |2 \vec{a}-3 \vec{b}|=10$
(2) When $|\vec{a}|=2, |\vec{b}|=1$ and $\vec{a}-3 \vec{b}$ is perpendicular to $2 \vec{a}+\vec{b}$

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In space, there are points \( \mathrm{A}(ec{a}) \) and \( \mathrm{B}(ec{b})\). Find the position vector of the point that externally divides the line segment AB in the ratio m:n.

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Given the components of the vectors \( ec{a}=(3,-4), ec{b}=(-2,1) \), express the following vectors in component form.
(1) 2 ec{a}
(2) -ec{b}
(3) ec{a}+2 ec{b}
(4) 2 ec{a}-3 ec{b}

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The existence range of point $\mathrm{P}$ that satisfies the vector equation: $\overrightarrow{\mathrm{OP}}=s \overrightarrow{\mathrm{OA}}+t \overrightarrow{\mathrm{OB}}$

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(4) Given \( \vec{a}=(3,5,-8), \vec{b}=(2,4,-6) \) and a real number $t$, let \( \vec{p}=(1-t) \vec{a}+t \vec{b} \). Find the value of $t$ that minimizes $|\vec{p}|$ and the value of $|\vec{p}|$ at that point.

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For the vectors shown in the right diagram, list all pairs of vector numbers that meet the following criteria:
(1) Vectors with equal magnitude
(2) Vectors with the same direction
(3) Equal vectors
(4) Opposite vectors

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Understand the geometric meaning of vector addition and tackle Example 2!

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There are fixed points O, A, and a moving point P. Given \overrightarrow{\mathrm{OA}}=ec{a} and \overrightarrow{\mathrm{OP}}=ec{p} , when |6 ec{p}-3 ec{a}|=2 , point P lies on the circumference of a certain circle. Find the center and radius of that circle. Assume ec{a}
eq \overrightarrow{0} .

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In the quadrilateral $A B C D$ shown on the right, which is a rhombus, point $O$ is the intersection of the diagonals $A C$ and $\mathrm{BD}$. Given that $\overrightarrow{\mathrm{OA}}=\vec{a}, \overrightarrow{\mathrm{AB}}=\vec{b}, \overrightarrow{\mathrm{CD}}=\vec{c}$,
(1) Draw the vectors $\vec{a}-\vec{b}$ and $\vec{a}-\vec{c}$. (2) What kind of vector is $\vec{b}+\vec{c}$?

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Decomposition of a vector ec{a}
eq \overrightarrow{0}, ec{b}
eq \overrightarrow{0}, ec{a} imes ec{b}. Any vector ec{p} can be uniquely expressed as ec{p}=s ec{a}+t ec{b} using real numbers $s, t$.

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In a coordinate space, find the distance between two points. If point A has coordinates (a1, a2, a3) and point B has coordinates (b1, b2, b3), what is the distance between A and B?

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Let's think about the geometric meaning of vector addition.
For the vectors $\vec{a}, \vec{b}$ on the right, draw $\vec{a}+\vec{b}$。

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Determine the value of $x$ such that the following two vectors ec{a}, ec{b} are parallel. (1) \( ec{a}=(3, x), ec{b}=(1,4) \) (2) \( ec{a}=(2 x, 9), ec{b}=(8, x) \)

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In $riangle \mathrm{ABC}$, there is a point $\mathrm{P}$ inside such that $2 \overrightarrow{\mathrm{PA}}+3 \overrightarrow{\mathrm{PB}}+5 \overrightarrow{\mathrm{PC}}=\overrightarrow{0}$ holds true.
(1) Where is point $\mathrm{P}$ located?
(2) Find the area ratio $riangle \mathrm{PBC}: riangle \mathrm{PCA}: riangle \mathrm{PAB}$.

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Example Question 35 Minimum Magnitude of Vector (Space) Let \(ec{a}=(2,-4,-3)\) and \(ec{b}=(1,-1,1)\). Find the minimum magnitude of ec{a}+t ec{b} (where $t$ is a real number) and the value of $t$ at that minimum. [Chiba Institute of Technology]

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The scalar multiple of a vector\ Given a real number $k$ and a vector \( \vec{a}(\neq \overrightarrow{0}) \), the $k$-multiple of vector $\vec{a}$, denoted as $k \vec{a}$, is defined as follows:
1. If $k > 0$, the vector $k \vec{a}$ has the same direction as $\vec{a}$, but its magnitude is $k$ times. Specifically, $1 \vec{a}=\vec{a}$
2. If $k < 0$, the vector $k \vec{a}$ has the opposite direction of $\vec{a}$, and its magnitude is $|k|$ times. Specifically, \( \quad (-1) \vec{a}=-\vec{a} \)
3. If $k=0$, the result is the zero vector $\overrightarrow{0}$, meaning $0 \vec{a}=\overrightarrow{0}$
Next, verify this with the given example. Example: Multiply the vector \( \vec{a} = (3, -2) \) by the real numbers $k = 2, -3, 0$.

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Alternative solution method for $\overrightarrow{\mathrm{DH}}$ (same up to step 11). With point D as the starting point,
$\vec{b} = \overrightarrow{\mathrm{DB}} - \overrightarrow{\mathrm{DA}}, \quad \vec{c} = \overrightarrow{\mathrm{DC}} - \overrightarrow{\mathrm{DA}}, \quad \vec{d} = -\overrightarrow{\mathrm{DA}}$
Therefore, from (1),
\[ egin{aligned} \overrightarrow{\mathrm{DH}} & = \frac{1}{30} k (\overrightarrow{\mathrm{DB}} - \overrightarrow{\mathrm{DA}}) + \frac{1}{5} k (\overrightarrow{\mathrm{DC}} - \overrightarrow{\mathrm{DA}}) - \frac{9}{10} k (-\overrightarrow{\mathrm{DA}})
& = \frac{2}{3} k \overrightarrow{\mathrm{DA}} + \frac{1}{30} k \overrightarrow{\mathrm{DB}} + \frac{1}{5} k \overrightarrow{\mathrm{DC}} \end{aligned} \]
$\leqslant \frac{1}{10} + \frac{3}{5} + \frac{3}{10} = 1$
Check this.

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Given z=4+2i, lpha=1+3i , plot the points \( \mathrm{P}(z), \mathrm{A}(lpha), \mathrm{A}^{\prime}(-lpha), \mathrm{B}(z+lpha), \mathrm{C}(z-lpha) \) on the complex plane.

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The coordinates of point A are (2,-4), the coordinates of point B are (-2,2), and the coordinates of point C are (0,-4). For the vectors ec{a}, ec{b}, ec{c}, please answer the following questions:
(1) Represent the vectors ec{a}, ec{b}, ec{c} in component form.
(2) Calculate the magnitudes of |ec{a}|,|ec{b}|,|ec{c}|.

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Prove that the following equations hold. (1) $\overrightarrow{\mathrm{PQ}}+\overrightarrow{\mathrm{RS}}+\overrightarrow{\mathrm{QR}}+\overrightarrow{\mathrm{SP}}=\overrightarrow{0}$ (2) $\overrightarrow{\mathrm{PB}}+\overrightarrow{\mathrm{DS}}-\overrightarrow{\mathrm{PS}}-\overrightarrow{\mathrm{XB}}=\overrightarrow{\mathrm{DX}}$

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Position vector, application to shapes Position vector of dividing points (space)

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Find the equation of the plane that passes through the point \( \mathrm{A}(2,1,-5) \) and is perpendicular to the vector \( ec{n}=(1,-2,3) \).

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Please solve the problem related to vectors in space.

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In each of the following situations, find the angle $\theta$ between $\vec{a}$ and $\vec{b}$.
(1) When $|\vec{a}|=2,|\vec{b}|=3,|2 \vec{a}+\vec{b}|=\sqrt{13}$
(2) When $|\vec{a}|=2,|\vec{b}|=\sqrt{3}$ and $\vec{a}-\vec{b}$ is perpendicular to $6 \vec{a}+7 \vec{b}$

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On a plane, consider △ABC and points P and Q. Answer the positions of points P and Q when the following equations hold:
(1) 3 \overrightarrow{AP} - \overrightarrow{AB} - 2 \overrightarrow{AC} = \overrightarrow{0}
(2) 4 \overrightarrow{AQ} + \overrightarrow{BQ} + 2 \overrightarrow{CQ} = \overrightarrow{0}

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Find the centroid coordinates of points A, B, and C. If the coordinates of point A are (a1, a2, a3), the coordinates of point B are (b1, b2, b3), and the coordinates of point C are (c1, c2, c3), what are the centroid coordinates?

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For two points \( \mathrm{A}(a_1, a_2) \) and \( \mathrm{B}(b_1, b_2) \), \[ \overrightarrow{\mathrm{AB}} = \left(b_1 - a_1, b_2 - a_2
ight) \] \[ |\overrightarrow{\mathrm{AB}}| = \sqrt{(b_1 - a_1)^2 + (b_2 - a_2)^2} \]

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In the triangle $riangle \mathrm{ABC}$ with vertices at points \( \mathrm{A}(ec{a}), \mathrm{B}(ec{b}), \mathrm{C}(ec{c}) \), let $\mathrm{P}$ be the midpoint of segment $\mathrm{AB}$, $\mathrm{Q}$ the point that externally divides segment $\mathrm{BC}$ in the ratio 1:2, and $\mathrm{R}$ the point that externally divides segment $\mathrm{CA}$ in the ratio 2:1. Let G be the centroid of $riangle \mathrm{AQR}$. Express the following vectors using ec{a}, ec{b}, ec{c} :
(1) The position vector of point G
(2) $\overrightarrow{\mathrm{PG}}$

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If point P is on the plane KLM, find the magnitude of vector OP |€{ec{\mathrm{OP}}}|.

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Find the equation of the plane that passes through the point \( \mathrm{A}(4,2,2) \) and is perpendicular to the vector \( ec{n}=(2,-3,1) )。

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For $riangle \mathrm{OAB}$, let $\overrightarrow{\mathrm{OP}}=s \overrightarrow{\mathrm{OA}}+t \overrightarrow{\mathrm{OB}}$. Given that real numbers $s$ and $t$ satisfy s+t=rac{1}{3}, s \geqq 0, t \geqq 0 , find the range of existence of point $\mathrm{P}$.

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Given \(ec{a}=(3,-4)\) and \(ec{b}=(-2,1)\), express the components of the following vectors:
(1) 2ec{a}
(2) -ec{b}
(3) ec{a}+2ec{b}

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For triangle OAB, let $\overrightarrow{\mathrm{OP}}=s \overrightarrow{\mathrm{OA}}+t \overrightarrow{\mathrm{OB}}$. Determine the range of existence for point P when the real numbers $s, t$ satisfy the following equations. (1) $s+t=2$ (2) $s+t=3, \quad s \geqq 0, \quad t \geqq 0$

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Chapter 1 Vectors on the Plane- 23
EX 3 points \( \mathrm{A}(1,1), \mathrm{B}(3,2), \mathrm{C}(5,-2) \) exist.
(1) Find the cosine of the angle $\theta$ between $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{AC}}$, which is $\cos \theta$.
(2) Find the area of the triangle $\triangle \mathrm{ABC}$.
(3) Find the real number t that minimizes the magnitude of the vector $t \overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{AC}}$ and its minimum value.

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Regarding the vectors shown in the diagram on the right, list all pairs of vector numbers as follows:
(1) Vectors with equal magnitude
(2) Vectors with the same direction
(3) Equal vectors
(4) Opposite vectors

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Given \( ec{a}=(1,2,3), ec{b}=(2,0,-1) \), find the minimum value of |ec{c}| and the value of $t$ at that minimum, where ec{c}=ec{a}+t ec{b} .

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Suppose in the plane, there are $riangle \mathrm{ABC}$ and points $\mathrm{P}$ and $\mathrm{Q}$. When the following equations hold, answer the positions of points $\mathrm{P}$ and $\mathrm{Q}$.
(1) $5 \overrightarrow{\mathrm{AP}}-2 \overrightarrow{\mathrm{AB}}-3 \overrightarrow{\mathrm{AC}}=\overrightarrow{0}$
(2) $7 \overrightarrow{\mathrm{AQ}}+2 \overrightarrow{\mathrm{BQ}}+3 \overrightarrow{\mathrm{CQ}}=\overrightarrow{0}$

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Given vectors \vec{a} and \vec{b}. If |\vec{a}| = 2\sqrt{10}, |\vec{b}| = \sqrt{5}, and \vec{a} \cdot \vec{b} = -10, answer the following questions. (1) For any real number t, find the minimum value of |\vec{a} + t\vec{b}| and the value of t at that time. (2) For the value of t obtained in (1), prove that \vec{a} + t\vec{b} is perpendicular to \vec{b}.

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In space, a directed line segment from point A to point B, represented by $\overrightarrow{\mathrm{AB}}$, has a magnitude represented by $|\overrightarrow{\mathrm{AB}}|$. Space vectors are also often represented by lowercase letters like ec{a} and ec{b} . Space vectors are defined in exactly the same way as plane vectors. Please answer the following questions about the basic properties of vectors.
1. If ec{a} and ec{b} have the same direction and magnitude, how can they be represented?
2. How is the inverse vector of ec{a} represented?
3. What are the vectors with a magnitude of 0 and a magnitude of 1 called, respectively?
4. Give examples of vector addition, subtraction, and scalar multiplication.

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Given z=3+2 i, lpha=1-i , plot the points \( \mathrm{P}(z), \mathrm{A}(lpha), \mathrm{P}^{\prime}(-z), \mathrm{B}(z+lpha), \mathrm{C}(z-lpha) \) on the complex plane.

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(1) Find the value of $x$ such that \( \vec{a}=(x+2,1) \) and \( \vec{b}=(1,-6) \) are perpendicular.
(2) Find the vector $\vec{d}$, which is perpendicular to \( \vec{c}=(2,1) \) and has a magnitude of $2 \sqrt{5}$.

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Find the equation of a line that satisfies the following conditions using vectors:
(1) Passes through point \( \mathrm{A}(-2,3) \) and is parallel to vector \( ec{d}=(2,1) \)
(2) Passes through two points \( \mathrm{A}(-1,2) \) and \( \mathrm{B}(3,1) \)

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For the following vectors \vec{a}, \vec{b}, illustrate \vec{a}-\vec{b}.

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(1) In triangle $riangle \mathrm{OAB}$, given \overrightarrow{\mathrm{OA}}=ec{a}, \overrightarrow{\mathrm{OB}}=ec{b} , express the area $S$ of $riangle \mathrm{OAB}$ using ec{a} and ec{b} .
(2) Using (1), when $|\overrightarrow{\mathrm{OA}}|=3, |\overrightarrow{\mathrm{OB}}|=4, \overrightarrow{\mathrm{OA}} \cdot \overrightarrow{\mathrm{OB}}=-6$, find the area $S$ of $riangle \mathrm{OAB}$.

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Find the coordinates of the point P that internally divides the line segment AB in the ratio m:n. Given that the coordinates of point A are (a1, a2, a3) and the coordinates of point B are (b1, b2, b3), what are the coordinates of point P?

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The coordinates of the midpoint of line segment AB
$\ \ left(\rac{a_{1}+b_{1}}{2}, \rac{a_{2}+b_{2}}{2}, \rac{a_{3}+b_{3}}{2}$

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(1) $3 \overrightarrow{\mathrm{AP}}-\overrightarrow{\mathrm{AB}}-2 \overrightarrow{\mathrm{AC}}=\overrightarrow{0}$

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When points O, P, and C are collinear in this order, P is the point closer to O among the intersection points of line OC and sphere S. Given that OC = √(0^2+1^2+2^2) = √5, what is the y-coordinate of point P when points O, P, and C are collinear in this order?

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Basic example
Addition of 2-dimensional vectors
For the vectors $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ in the right diagram, illustrate the following vectors.
(1) $\vec{a}+\vec{b}$
(2) $\vec{a}+\vec{b}+\vec{c}$
(3) $\vec{a}+\vec{d}$

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Conditions for points to match
Points $\mathrm{P}, \mathrm{Q}$ match
$\Longleftrightarrow \overrightarrow{\mathrm{OP}}=\overrightarrow{\mathrm{OQ}}$ (position vectors match)
\& refer to 50.

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In space, there are points \( \mathrm{A}(ec{a}), \mathrm{B}(ec{b}) \). Find the position vector of the point that divides the line segment AB in the ratio $m:n$.

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Prove that the following equations hold.
(1) $\overrightarrow{\mathrm{PQ}}+\overrightarrow{\mathrm{RS}}+\overrightarrow{\mathrm{QR}}+\overrightarrow{\mathrm{SP}}=\overrightarrow{0}$
(2) $\overrightarrow{\mathrm{PB}}+\overrightarrow{\mathrm{DS}}-\overrightarrow{\mathrm{PS}}-\overrightarrow{\mathrm{XB}}=\overrightarrow{\mathrm{DX}}$

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Join our Discord#### Q.22

Find the coordinates of the midpoint of line segment AB. If the coordinates of point A are (a1, a2, a3) and the coordinates of point B are (b1, b2, b3), what are the coordinates of the midpoint?

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Components of a vector Decomposition and components of a vector

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Find the vector $\vec{b}$ which makes an angle of $45^{\circ}$ with the vector \( \vec{a}=(1,2) \) and has a magnitude of $\sqrt{10}$.

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Find the angle between $\overrightarrow{\mathrm{OC}}$ and $\overrightarrow{\mathrm{MN}}$.

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Find the equation of the plane passing through the points \( \mathrm{A}(1,-1,0), \mathrm{B}(3,1,2), \mathrm{C}(3,3,0) \).

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In the regular hexagon $\mathrm{ABCDEF}$ in the right figure, let the intersection of diagonals $\mathrm{AD}$ and $\mathrm{BE}$ be $\mathrm{O}$, and let \overrightarrow{\mathrm{OA}} = ec{a}, \overrightarrow{\mathrm{OB}} = ec{b}. In this case, express the following vectors using ec{a} and ec{b}:
(1) $\overrightarrow{\mathrm{DE}}$
(2) $\overrightarrow{\mathrm{FC}}$
(3) $\overrightarrow{\mathrm{AC}}$
(4) $\overrightarrow{\mathrm{BF}}$

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Join our Discord#### Q.28

What kind of figure do the following parametric equations draw?
(1) $x=t+1, \quad y=3t-2$
(2) $x=\sqrt{t}-1, \quad y=t-3\sqrt{t}$