Calculus Notes 13.4

Calculus Notes 13.4 - Calculus-Stewart Dr Berg Summer 09...

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Calculus- Stewart Dr. Berg Summer ‘09 Page 1 13.4 13.4 The Cross Product The cross product or vector product is a rather peculiar construction since it is defined only for real three vectors, but it is a natural construction for things like torque. Definition If a = ( a 1 , a 2 , a 3 ) and b = ( b 1 , b 2 , b 3 ) then the cross product of a and b is the vector a × b = ( a 2 b 3 a 3 b 2 , a 3 b 1 a 1 b 3 , a 1 b 2 a 2 b 1 ) . Theorem If a = a 1 i + a 2 j + a 3 k and b = b 1 i + b 2 j + b 3 k then a × b = i j k a 1 a 2 a 3 b 1 b 2 b 3 = a 2 a 3 b 2 b 3 i a 1 a 3 b 1 b 3 j + a 1 a 2 b 1 b 2 k = ( a 2 b 3 a 3 b 2 ) i + ( a 1 b 3 a 3 b 1 ) j + ( a 1 b 2 a 2 b 1 ) k . Example A ( i 2 j + 3 k ) × (2 i + j k ) = 2 3 1 1 i + 1 3 2 1 j + 1 2 2 1 k = ( 2)( 1) (3)(1) [ ] i (1)( (3)(2) [ ] j + (1)(1) ( 2)(2) [ ] k = i + 7 j + 5 k Example B Given any vector a = ( a 1 , a 2 , a 3 ) , a × a = 0 . Indeed, a × a = a 2 a 3 a 2 a 3 i + a 1 a 3 a 1 a 3 j + a 1 a 2 a 1 a 2 k = 0 i + 0 j + 0 k = 0 . Theorem i) a × b is orthogonal to the plane α a + β b | , R { } generated by a and b .
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Calculus Notes 13.4 - Calculus-Stewart Dr Berg Summer 09...

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