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Extra resources for Algorithmic Geometry [auth. unkn.]
12. 6. Given points A, B, and C with position vectors a, b, and c. Suppose that c = ma + nb for some real numbers m and n, and that 0, A, B are not collinear, show that A, B, and Care collinear if and only if m + n = 1. 13) 7. Consider 6ABC. Let M be the mid-point of the median AD. Through M draw a straight line to meet AB at P and AC at Q. 21 Vectors, Matrices and Geometry (a) Suppose now A is placed at the origin O. Express the position vector of M as a linear combination of p and q. (b) Hence deduce that ~~ + ~g is a constant.
8. 7. Show that for any three vectors x, y, z of R3 and any scalars r, s, t, the three vectors rx - sy, ty - rz, sz - tx are linearly dependent. 8. Given three linearly independent vectors x, y, z we construct the vectors a = 2x + y, b = y - z and c = x + y + z. Express the vector d = x + 2y + z as a linear combination of a, band c. 4 Geometry in space In this section we shall use the one-to-one correspondence be- 52 Vectors and Geometry in Space tween points in space and their positional vectors to study geometry in space.
V12 2 + 42) = 40/(J20· V160) = 1/V2 . Therefore L1 and L2 intersect each other at an angle of 45°. 36 Vectors and Geometry in the Plane EXERCISES 1. In what follows, find a vector equation for the straight line containing the given point P and having the given vector a as a normal vector. (a) P = (b) P = (c) P = (d) P = (-1,2), a = [1, -2J. (3,2), a = [1, OJ. (-4,1)' a = [5,2]. (-3, -2), a = [2,3]. Hence express the equations in the form ax + by + c = o. 2. In each of the following, find a vector equation for the straight line containing the given point P and perpendicular to the line passing through given points A and B.