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13.3Problems57-67

# 13.3Problems57-67 - 338 C H A P T E R 13 V E C T O R G E O...

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338 C H A P T E R 13 VECTOR GEOMETRY (ET CHAPTER 12) y x e 1 = 1, 0 e 2 = , 2 2 2 2 (b) Notice from the picture that if we rotate e 1 by π / 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 1 = 2 2 , 2 2 and e 2 = 0 , 1 . Note that e 1 · e 2 = 1 · 2 2 + 0 · 2 2 = 2 2 and e 1 · e 2 = 0 · 2 2 + 1 · 2 2 = 2 2 . Thus, e 1 · e 2 = e 1 · e 2 . 57. Determine v + w if v and w are unit vectors separated by an angle of 30 . SOLUTION We use the relation of the dot product with length and properties of the dot product to write v + w 2 = ( v + w ) · ( v + w ) = v · v + v · w + w · v + w · w = v 2 + 2 v · w + w 2 = 1 2 + 2 v · w + 1 2 = 2 + 2 v · w We now find v · w : v · w = v w cos 30 = 1 · 1 cos 30 = 3 2 Hence, v + w 2 = 2 + 2 · 3 2 = 2 + 3 v + w = 2 + 3 1 . 93 58. What is the angle between v and w if: (a) v · w = − v w (b) v · w = 1 2 v w SOLUTION (a) Since v · w = v w cos θ , it follows that v w cos θ = − v v cos θ = − 1 The solution for 0 θ 180 is θ = 180 . (b) By v · w = v w cos θ we get v w cos θ = 1 2 v w cos θ = 1 2 The solution for 0 θ 180 is θ = 60 . 59. Suppose that v = 2 and w = 3, and the angle between v and w is 120 . Determine: (a) v · w (b) 2 v + w (c) 2 v 3 w SOLUTION (a) We use the relation between the dot product and the angle between two vectors to write v · w = v w cos θ = 2 · 3 cos 120 = 6 · 1 2 = − 3 (b) By the relation of the dot product with length and by properties of the dot product we have 2 v + w 2 = ( 2 v + w ) · ( 2 v + w ) = 4 v · v + 2 v · w + 2 w · v + w · w = 4 v 2 + 4 v · w + w 2 We now substitute v · w = − 3 from part (a) and the given information, obtaining 2 v + w 2 = 4 · 2 2 + 4 ( 3 ) + 3 2 = 13 2 v + w = 13 3 . 61

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S E C T I O N 13.3 Dot Product and the Angle Between Two Vectors (ET Section 12.3) 339 (c) We express the length in terms of a dot product and use properties of the dot product. This gives 2 v 3 w 2 = ( 2 v 3 w ) · ( 2 v 3 w ) = 4 v · v 6 v · w 6 w · v + 9 w · w = 4 v 2
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13.3Problems57-67 - 338 C H A P T E R 13 V E C T O R G E O...

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