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Let b = 2 i + j + k , R . If a is a vector such that a b = -7 i + 8 j - 3 k and a b = -2 , then for c = 2 i + 3 j - k , the value of a ( c ( a b )) is equal to

Correct answer

136

Step-by-step solution

Since a b is orthogonal to b , we have b ( a b ) = 0 . 2(-7) + 1(8) + (-3) = 0 -14 + 8 - 3 = 0 = -2 Thus, b = 2 i + j - 2 k and | b |^2 = 4 + 1 + 4 = 9 . We know the vector triple product identity: b ( a b ) = ( b b ) a - ( b a ) b | b |^2 a = b ( a b ) + ( a b ) b First, compute b ( a b ) : b ( a b ) = vmatrix i & j & k 2 & 1 & -2 -7 & 8 & -3 vmatrix = 13 i + 20 j + 23 k Next, compute ( a b ) b = -2(2 i + j - 2 k ) = -4 i - 2 j + 4 k Substitute these into the identity: 9 a = (13 i + 20 j + 23 k ) + (-4 i - 2 j + 4

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