p0131

# p0131 - 1.31 CHAPTER 1 PROBLEM 31 33 1.31 Chapter 1 Problem...

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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

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

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