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 CHAPTER 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 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 . SOLUTION 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 =k 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 (a) Since v · w v kk w k cos θ , it follows that k v kk w k cos 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 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 (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) By the relation of the dot product with length and by properties of the dot product we have k 2 v + w k 2 = ( 2 v + w ) · ( 2 v + w ) = 4 v · v + 2 v · w + 2 w · v + w · w = 4 k v k 2 + 4 v · w w k 2 We now substitute v · w 3 from part (a) and the given information, obtaining k 2 v + w k 2 = 4 · 2 2 + 4 ( 3 ) + 3 2 = 13 2 v + w 13 3 . 61
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SECTION 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 k 2 v 3 w k 2 = ( 2 v 3 w ) · ( 2 v 3 w ) = 4 v · v 6 v · w 6 w · v + 9 w · w = 4 k v k 2 12 v · w + 9 k w k 2 Substituting v · w =−
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This note was uploaded on 02/13/2010 for the course MATH MATH 32A taught by Professor Park during the Fall '09 term at UCLA.

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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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