654 CHAPTER 9 Polar Coordinates; Vectors 4 Determine Whether Two Vectors Are Orthogonal If the angle θ between two nonzero vectors v and w is 2 , π the vectors v and w are called orthogonal . See Figure 70. Since cos 2 0, π = it follows from formula (8) that if the vectors v and w are orthogonal, then v w 0. ⋅ = On the other hand, if v w 0, ⋅ = then v 0 = or w 0 = or cos 0. θ = If cos 0, θ = then 2 , θ π = and v and w are orthogonal. If v or w is the zero vector, then, since the zero vector has no specific direction, we adopt the convention that the zero vector is orthogonal to every vector. Determining Whether Two Vectors Are Parallel The vectors v i j 3 = − and w i j 6 2 = − are parallel, since v w 1 2 . = Furthermore, since EXAMPLE 3 v w v w cos 18 2 10 40 20 400 1 θ = ⋅ = + = = the angle θ between v and w is 0. Figure 70 v is orthogonal to w . v w NOTE Orthogonal, perpendicular , and normal are all terms that mean “meet at a right angle.” It is customary to refer to two vectors as being orthogonal , to two lines as being perpendicular , and to a line and a plane or a vector and a plane as being normal . ■ Determining Whether Two Vectors Are Orthogonal The vectors v i j w i j 2 and 3 6 = − = + are orthogonal, since v w 6 6 0 ⋅ = − = See Figure 71. EXAMPLE 4 THEOREM Two vectors v and w are orthogonal if and only if v w 0 ⋅ = Now Work PROBLEM 9(C) 5 Decompose a Vector into Two Orthogonal Vectors In many physical applications, it is necessary to find “how much” of a vector is applied in a given direction. Look at Figure 72. The force F due to gravity is pulling straight down (toward the center of Earth) on the block. To study the effect of gravity on the block, it is necessary to determine how much of F is actually pushing the block down the incline F1 ( ) and how much is pressing the block against the incline F , 2 ( ) at a right angle to the incline. Knowing the decomposition of F often enables us to determine when friction (the force holding the block in place on the incline) is overcome and the block will slide down the incline. Suppose that v and w are two nonzero, nonorthogonal vectors with the same initial point P . We seek to decompose v into two vectors: v ,1 which is parallel to w , Figure 71 x y w 5 3i 1 6j v 5 2i 2 j Figure 72 F1 F2 F
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