By Kam-Tim Leung, S.N. Suen

This e-book introduces the $64000 thoughts of finite-dimensional vector areas in the course of the cautious research of Euclidean geometry. In flip, tools of linear algebra are then utilized in the examine of coordinate ameliorations during which an entire class of conic sections and quadric surfaces is received.

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5 EXAMPLE Let L1 and L2 be given by 4x - 2y + 7 =0 and 12x + 4y - 5 =0 . Then the angle () between them is given by cos() = (4 ·12 - 2· 4)/(V42 + 22. 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.

3) lira + sb 0 for scalars r and s then r = = = = = = 47 Vectors, Matrices and Geometry (4) a l b2 - a2bl # 0, a2b3 - a3b2 ::j:. °or a3bl - a l b3 ::j:. 0. 3 give complete information on the linear dependence and the linear independence of two vectors. In each of them condition (1) is the definition, (2) is a geometric characterization, (3) is given in terms of linear combination and (4) is a condition on the components of the vectors. Let us now look at the case of three vectors. First of all the defini tion.

A) Express the position vectors of P, Q, and R in terms of position vectors of AI, A 2, B l , B 2, Cl and C2. 13, show that P, Q, and Rare collinear. 11. Consider parallelogram ABCD. M and N are mid-points of BC and CD respectively. AM and AN meet the diagonal BD at E and F respectively. 5 The dot product In the last section we have successfully employed vector algebra in the study of plane geometry. In this section we shall study the algebraic notion and the geometric meaning of dot product in the vector space R2, which will be used in the next section to provide us with a link to the coordinate plane geometry.

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