13.3Problems57-67

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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 =−
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 5

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online