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Solution
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First, consider two vectors a and b that have the same direction angle q but different magnitudes ra and rb, so that a = (q,ra) and b = (q,rb) The magnitude of a b is ra b = rarb sin (q q) = rarb sin 0 = rarb 0 = 0 Whenever a vector has a magnitude of zero, then it s the zero vector by definition, so a b=0
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Now look at the case where a and b have angles that differ by p, so they point in opposite directions As before, you can assign the coordinates a = (q,ra) Two possibilities exist for the direction angle of a You can have 0 q<p or p q < 2p If 0 q < p, then b = [(q + p),rb] and the magnitude of a b is ra b = rarb sin [(q + p) q] = rarb sin p = rarb 0 = 0 Therefore a b=0 If p q < 2p, then b = [(q p),rb] In this case, the magnitude of a b is ra b = rarb sin [(q p) q] = rarb sin ( p) = rarb 0 = 0 so again, a b=0
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Here s an extra credit challenge!
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Prove that the cross product of two vectors is anticommutative That is, show that for any two polar-plane vectors a and b, the magnitudes of a b and b a are the same, but they point in opposite directions
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Solution
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You re on your own That s what makes this is an extra credit problem!
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Practice Exercises
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This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App A The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Consider two standard-form vectors a and b in the Cartesian plane, represented by the ordered pairs a = (5, 5) and b = ( 5,5) Calculate and compare the Cartesian products 4a and 4b 2 Convert the original two vectors from Problem 1 into polar form Then calculate and compare the polar products 4a and 4b 3 Prove that the multiplication of a standard-form vector by a positive scalar is right-hand distributive over Cartesian-plane vector subtraction That is, if k+ is a positive constant, and if a and b are vectors in the xy plane, then (a b)k+ = ak+ bk+ 4 Consider two standard-form vectors a and b in the Cartesian plane, represented by
a = (4,4)
and b = ( 7,7) Calculate and compare the Cartesian dot products a b and b a 5 Convert the original two vectors from Problem 4 into polar form Then calculate and compare the polar dot products a b and b a 6 Prove that the dot product is commutative for standard-form vectors in the Cartesian plane 7 Prove that the dot product is commutative for vectors in the polar plane 8 Prove that if k+ is a positive constant, and if a and b are standard-form vectors in Cartesian or polar two-space, then k+a k+b = k+2(a b) Demonstrate the Cartesian case first, and then the polar case
Practice Exercises
9 Consider the two polar vectors a = (p /3,4) and b = (3p /2,1) Determine the polar cross product a b 10 Consider the two polar vectors a = (p,8) and b = (7p /6,5) Determine the polar cross product a b