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p q A A A2 A2 A2 . 1 2 3
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THE CROSS OR VECTOR PRODUCT 7.16. Prove A B B A.
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A B C has magnitude AB sin  and direction such that A, B, and C form a right-handed system [Fig. 7-24(a)]. B A D has magnitude BA sin  and direction such that B, A, and D form a right-handed system [Fig. 7-24(b)]. Then D has the same magnitude as C but is opposite in direction, i.e., C D or A B B A. The commutative law for cross products is not valid.
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7.17. Prove that A B C A B A C for the case where A is perpendicular to B and also to C.
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VECTORS
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Fig. 7-24
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Since A is perpendicular to B, A B is a vector perpendicular to the plane of A and B and having magnitude AB sin 908 AB or magnitude of AB. This is equivalent to multiplying vector B by A and rotating the resultant vector through 908 to the position shown in Fig. 7-25.
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Fig. 7-25
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Fig. 7-26
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Similarly, A C is the vector obtained by multiplying C by A and rotating the resultant vector through 908 to the position shown. In like manner, A B C is the vector obtained by multiplying B C by A and rotating the resultant vector through 908 to the position shown. Since A B C is the diagonal of the parallelogram with A B and A C as sides, we have A B C A B A C.
7.18. Prove that A B C A B A C in the general case where A, B, and C are noncoplanar. See Fig. 7-26.
Resolve B into two component vectors, one perpendicular to A and the other parallel to A, and denote them by B and Bk respectively. Then B B Bk . If  is the angle between A and B, then B B sin . Thus the magnitude of A B is AB sin , the same as the magnitude of A B. Also, the direction of A B is the same as the direction of A B. Hence A B A B. Similarly, if C is resolved into two component vectors Ck and C , parallel and perpendicular respectively to A, then A C A C. Also, since B C B Bk C Ck B C Bk Ck it follows that A B C A B C Now B and C are vectors perpendicular to A and so by Problem 7.17, A B C A B A C A B C A B A C
Then
VECTORS
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and the distributive law holds. Multiplying by 1, using Problem 7.16, this becomes B C A B A C A. Note that the order of factors in cross products is important. The usual laws of algebra apply only if proper order is maintained.
  i  7.19. (a) If A A1 i A2 j A3 k and B B1 i B2 j B3 k, prove that A B  A1   B1
j A2 B2
 k   A 3 .  B3 
A B A1 i A2 j A3 k B1 i B2 j B3 k A1 i B1 i B2 j B3 k A2 j B1 i B2 j B3 k A3 k B1 i B2 j B3 k A1 B1 i i A1 B2 i j A1 B3 i k A2 B1 j i A2 B2 j j A2 B3 j k A3 B1 k i A3 B2 k j A3 B3 k k    i j k      A2 B3 A3 B2 i A3 B1 A1 B3 j A1 B2 A2 B1 k  A1 A2 A3     B1 B2 B3  (b) Use the determinant representation to prove the result of Problem 7.18.
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