p0131 - 1.31. CHAPTER 1, PROBLEM 31 33 1.31 Chapter 1,...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
1.31. CHAPTER 1, PROBLEM 31 33 1.31 Chapter 1, Problem 31 Problem: Because sin θ is multivalued while cos θ is single valued in the principal-value range, 0 θ π , using the cross product to determine the angle between two vectors is less reliable than using the dot product. Demonstrate this algebraically and graphically for the following two pairs of vectors. (a) a =3 i +4 j and b = i j . (b) a i j and b = i j . Solution: Thed i f fe rencebe tweenPa r ts(a )and(b )isthevec to r b . The figures below show the vectors involved in the cross product of a and b . xx yy zz a i ja i j b = i j a × b = 7 k b = i j a × b = k Part (a) Part (b) ................................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 . 1 o 171 . 9 o (a) From the definition of the cross product, a × b = n | a || b | sin θ For a i j and b = i j , the right-hand rule tells us that the unit normal is n = k . Also, the cross product of a and b is a × b = e e e e e e e ijk 340 1 10 e e e e e e e =(0 0) i (0 0) j +( 3 4) k = 7 k Therefore, the angle θ is given by 7 k = k (5)( 2) sin θ = θ =sin 1 w 7 5 2 W 1 X 7 2 10 ~ For the standard principal angle range, viz., 0 o θ 180 o , we conclude that θ =81 . 9 o or θ =98 . 1 o Now, we consider the dot product, which yields a unique value for θ as follows. θ =cos 1 w a · b | a || b | W 1 w 1 (5)( 2) W 1 X 2 10 ~ . 1 o
Background image of page 1

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

View Full DocumentRight Arrow Icon
34 CHAPTER 1. INTRODUCTION (b) For a =3 i +4 j and b = i j , the right-hand rule tells us that the unit normal is n = k . Also, the cross product of a and b is a × b = e e e e e e e ijk 340 1 10 e e e e e e e =(0 0) i (0 0) j +( 3+4) k = k Therefore, the angle θ is given by k = k (5)( 2) sin θ = θ =sin 1 w 1 5 2 W 1 X 2 10 ~ For the standard principal angle range, viz., 0 o θ 180 o , we conclude that θ =8 . 1 o or θ = 171 . 9 o Now, we consider the dot product, which yields a unique value for θ as follows.
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

p0131 - 1.31. CHAPTER 1, PROBLEM 31 33 1.31 Chapter 1,...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online