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Question: If \(a,b,c\) are vectors such that \(\lbrack abc\rbrack = 4\), then \(\lbrack a \times bb \times cc...

If a,b,ca,b,c are vectors such that [abc]=4\lbrack abc\rbrack = 4, then

[a×bb×cc×a]\lbrack a \times bb \times cc \times a\rbrack =

A

16

B

64

C

4

D

8

Answer

16

Explanation

Solution

[a×bb×cc×a]=(a×b).[(b×c)×(c×a)]\lbrack\mathbf{a} \times \mathbf{bb} \times \mathbf{cc} \times \mathbf{a}\rbrack = (\mathbf{a} \times \mathbf{b}).\lbrack(\mathbf{b} \times \mathbf{c}) \times (\mathbf{c} \times \mathbf{a})\rbrack

=(a×b).([bca]c−[bcc]a)=(a×b).([bca]c−0)= (\mathbf{a} \times \mathbf{b}).(\lbrack\mathbf{bca}\rbrack\mathbf{c} - \lbrack\mathbf{bcc}\rbrack\mathbf{a}) = (\mathbf{a} \times \mathbf{b}).(\lbrack\mathbf{bca}\rbrack\mathbf{c} - 0)

=[bca][abc]=[abc][abc]=4.4=16.= \lbrack\mathbf{bca}\rbrack\lbrack\mathbf{abc}\rbrack = \lbrack\mathbf{abc}\rbrack\lbrack\mathbf{abc}\rbrack = 4.4 = 16.