coordinates. such as Cartesian coordinates, whose "natural

Chapter 9, Problem 9.2

(choose chapter or problem)

coordinates. such as Cartesian coordinates, whose "natural basis" (9). Sec. 9.1. consists of three 0l1hogonal unit vectors. (a) Show that a = [2. -2.4]. b = [0.8.4], c = [-20. -4. 8] are orthogonaL (b) For what values of al are a = [aI- 2, 0] and b = [3.4, -IJ 0l1hogonal? (c) Show that the straight lines 4x + 2y = 1 and 5x - 10)" = 7 are orthogonaL (d) Find all unit vectors a = [a lo a2] in the plane orthogonal to [4, 31. (e) Find all vectors orthogonal to a = [2. l. 0]. Do they fonn a vector space? (I) For what c are the plane!> 4x - 2)" + 3.;; = 6 and 2x - cy + 5::: = 1 orthogonal? (g) Under what condition will the diagonals of a parallelogram be orthogonal? (Prove your answer.) (h) What is the angle between a light ray and its reflection in three orthogonal plane minors (known as a "corner reflector")"? (i) Discuss further applications in physics and geometry in which orthogonality plays a role.