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Question

If a,b,c are three non-zero vectors such that b is not perpendicular to both a and c and (a×b)×c=a×(b×c), then

A
a and c are always non-collinear
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B
a and c are always collinear
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C
a and c are always perpendicular
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D
a,b and c are always non-coplanar
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Solution

The correct option is B a and c are always collinear
Given
(a×b)×c=a×(b×c)....(i)

From definition, we have
(a×b)×c=(ac)b(bc)a....(ii)
a×(b×c)=(ac)b(ab)c....(iii)

From equation (i),(ii) and (iii)
(ac)b(bc)a=(ac)b(ab)c
i.e., (bc)a=(ab)c
i.e., a=(bc)(ab)c
i.e., a=λc where λ=(bc)(ab) is a scalar

Hence a and c are always collinear.
The correct answer is Opt:[B]

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