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## Geometry and Measurement

### Vector Analysis Geometry of Curves and Surfaces - Geometric Transformations (Translation, Rotation, Scaling)

#### Q.01

"Given the point P'(4,-3), rotate it counterclockwise by -π/3 around the origin to obtain the point Q'(x', y'), find the coordinates of Q'."

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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 an angle of -π/3.'

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'Convert polar coordinates (r, θ) to Cartesian coordinates (x, y).'

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'Find the equation of the curve obtained by translating the curve F(x, y) = 0 parallel to the x-axis by p and parallel to the y-axis by q.'

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'In triangle ABC, with A(a), B(b), C(c), find the vector equations for the following lines: (1) Line passing through the midpoints of sides AB and AC (2) Perpendicular bisector of side BC'

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'(1) Find the coordinates of point B when point A(2,1) is rotated around the origin O by π/4.\n(2) Given that the coordinates of the point obtained by rotating point A(2,1) around point P by π/4 are (1-√2, -2+2√2). Find the coordinates of point P.'

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'For the point P (6, -4, 2), find the coordinates of the following points.'

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'Translation of the given text into multiple languages.'

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'Find the coordinates of point Q when point P(3,1) is rotated around the origin O by π/4.'

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'The parabola y = 1/2 x^2, when shifted parallel to the □ in the x-axis direction and □ in the y-axis direction, touches both the lines y = -x and y = 3x.'

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'Find the coordinates of point Q obtained by rotating point P(-2,3) around the origin by \ \\frac{5}{6} \\pi \.'

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'Find the coordinates of point Q after rotating point P(-2,3) around the origin by 5/6 π.'

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'(1) Translated 2 units along the x-axis and -7 units along the y-axis\n(2) a=3, b=1, c=-5'

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'Parametric representation of an ellipse'

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'What is the equation of the shape formed by moving a curve downwards by -2 units along the y-axis?'

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'Application of vectors to plane geometry'

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"(1) Find the coordinates of point B after rotating point A(2,1) around the origin by π/3. (2) Point Q's coordinates after rotating point A(2,1) around point P by π/3 were Q(3/2-3√3/2,-1/2+√3/2). Find the coordinates of point P."

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'Given a triangle ABC with vertices A(vector a), B(vector b), C(vector c). Find the vector equation of the line passing through the centroid G of triangle ABC and parallel to side AC.'

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'For triangle OAB, if OP=sOA+tOB, find the range of existence of point P when real numbers s and t satisfy the following conditions: (1) s+t=3 (2) 2s+3t=1, s≥0, t≥0 (3) 2s+3t≤6, s≥0, t≥0'

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'Plot the range of movement of point z on the complex plane when the complex number z satisfies |z-1| ≤ |z-4| ≤ 2|z-1|.'

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'Find the coordinates of point B after rotating point A(2,1) around the origin by π/3.'

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'Basic Example 88 Rotate a point around a point that is not the origin'

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'(1) Let point $\\mathrm{P}(X, Y)$ be rotated by an angle $\\theta$ around the origin to obtain point $\\mathrm{Q}(x, y)$. Express $X, Y$ in terms of $x, y, \\theta$. \n(2) Find the equation of the curve $5 x^{2}+2 \\sqrt{3} x y+7 y^{2}=16$ after rotating it around the origin by $\\frac{\\pi}{6}$.'

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'Given the point P(3,-2,1), find the coordinates of the following points:\n(1) The foot A of the perpendicular from point P to the xy plane\n(2) The point B which is symmetrical to point P with respect to the xy plane\n(3) The point C which is symmetrical to point P with respect to the z-axis\n(4) The point D which is symmetrical to point P with respect to the origin'

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'In tetrahedron ABCD, let E be the centroid of triangle BCD, and let G be the point that divides segment AE in the ratio 3:1. Prove that the equation AG+BG+CG+DG=0 holds.'

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Standard 69 | Translation of Parabola (1)

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Development 80 | Symmetric Transformation of Graphs

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What kind of movement does the point z-lpha represent?

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Describe the operation of rotating about a point other than the origin on the complex plane, and represent it using complex numbers. For example, show the case of rotating about the point $z = 1 + i$ by an angle of $\pi/4$ (45 degrees).