1211s1 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 1st Semester: Solution to Assignment 1 1. Let a = a 1 i + a 2 j + a 3 k , b = b 1 i + b 2 j + b 3 k and c = c 1 i + c 2 j + c 3 k . (a) We first prove that ( a × b ) × i = ( a · i ) b ( b · i ) a . Indeed, ( a × b ) × i = ± ± ± ± ± ± i j k a 1 a 2 a 3 b 1 b 2 b 3 ± ± ± ± ± ± × i = ± ± ± ± a 1 a 3 b 1 b 3 ± ± ± ± ( j × i ) + ± ± ± ± a 1 a 2 b 1 b 2 ± ± ± ± ( k × i ) = ( a 1 b 3 b 1 a 3 ) k + ( a 1 b 2 a 2 b 1 ) j = a 1 ( b 2 j + b 3 k ) b 1 ( a 2 j + a 3 k ) = ( a · i ) b ( b · i ) a . Similarly, we have ( a × b ) × j = ( a · j ) b ( b · j ) a and ( a × b ) × k = ( a · k ) b ( b · k ) a . Hence, ( a × b ) × c = c 1 [( a × b ) × i ] + c 2 [( a × b ) × j ] + c 3 [( a × b ) × k ] = c 1 [( a · i ) b ( b · i ) a ] + c 2 [( a · j ) b ( b · j ) a ] + c 3 [( a · k ) b ( b · k ) a ] = ( a · c ) b ( b · c ) a . (b) Using (a) and the identity
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1211s1 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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