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58.59.60.64in13.3

# 58.59.60.64in13.3 - S E C T I O N 13.3 Dot Product and the...

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SECTION 13.3 Dot Product and the Angle Between Two Vectors (ET Section 12.3) 337 The cosine of the angle α between e θ and the vector i in the direction of the positive x -axis is cos = e · i k e k·k i k = e · i = (( cos ) i + ( sin ) j ) · i = cos The solution of cos = cos for angles between 0 and 180 is = . That is, the vector e makes an angle with the x -axis. We now use the trigonometric identity cos cos ψ + sin sin = cos ( ) to obtain the following equality: e · e =h cos , sin i·h cos , sin i= cos cos + sin sin = cos ( ) 56. Let v and w be vectors in the plane. (a) Use Theorem 2 to explain why the dot product v · w does not change if both v and w are rotated by the same angle . (b) Sketch the vectors e 1 = h 1 , 0 i and e 2 = * 2 2 , 2 2 + , and determine the vectors e 0 1 , e 0 2 obtained by rotating e 1 , e 2 through an angle π 4 .Verifythat e 1 · e 2 = e 0 1 · e 0 2 . SOLUTION (a) By Theorem 2, v · w =k v kk w k cos where is the angle between v and w . Since rotation by an angle does not change the angle between the vectors, nor the norms of the vectors, the dot product v · w remains unchanged. y x e 1 = 1, 0 e 2 = , 2 2 2 2 (b) Notice from the picture that if we rotate e 1 by / 4, we get e 2 , and when we rotate e 2 by the same amount we get a unit vector along the y axis. Thus, e 0 1 = D 2 2 , 2 2 E and e 0 2 = h 0 , 1 i . Note that e 1 · e 2 = 1 · 2 2 + 0 · 2 2 = 2 2 and e 0 1 · e 0 2 = 0 · 2 2 + 1 · 2 2 = 2 2 . Thus, e 1 · e 2 = e 0 1 · e 0 2 . 57. Determine k v + w k if v and w are unit vectors separated by an angle of 30 . We use the relation of the dot product with length and properties of the dot product to write k v + w k 2 = ( v + w ) · ( v + w ) = v · v + v · w + w · v + w · w v k 2 + 2 v · w +k w k 2 = 1 2 + 2 v · w + 1 2 = 2 + 2 v · w We now Fnd v · w : v · w v kk w k cos 30 = 1 · 1 cos 30 = 3 2 Hence, k v + w k 2 = 2 + 2 · 3 2 = 2 + 3 ⇒k v + w k= q 2 + 3 1 . 93 58. What is the angle between v and w if: (a) v · w =−k v kk w k (b) v · w = 1 2 k v w k

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338 CHAPTER 13 VECTOR GEOMETRY (ET CHAPTER 12) (a) Since v · w =k v kk w k cos θ , it follows that k v kk w k cos =−k v kk v k cos =− 1 The solution for 0 180 is = 180 . (b) By v · w v kk w k cos we get k v kk w k cos = 1 2 k v kk w k cos = 1 2 The solution for 0 180 is = 60 . 59. Suppose that k v k= 2and k w 3, and the angle between v and w is 120 . Determine: (a) v · w( b ) k 2 v + w k (c) k 2 v 3 w k SOLUTION (a) We use the relation between the dot product and the angle between two vectors to write v · w v kk w k cos = 2 · 3 cos 120 = 6 · µ 1 2 3 (b)
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58.59.60.64in13.3 - S E C T I O N 13.3 Dot Product and the...

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