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Question

If b, c,d are non-coplanar vectors then the vector (a×b)×(c×d)+(a×c)×(d×b)+(a×d)×(b×c) is parallel to

A
a
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B
b
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C
c
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D
a+b
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Solution

The correct option is B a
Given expression

(a×b)×(c×d)+(a×c)×(d×b)+(a×d)×(b×c)
((a×b)d)c((a×b)c)d+((d×b)a)c((d×b)c)a+((b×c)a)d((b×c)d)a
[abd]c[abc]d+[abd]c[bcd]a+[abc]d[bcd]a
2[abd]c2[bcd]a
Let's check given expression is parallel to a
a×(2[abd]c2[bcd]a)=0

2a×a(b×d)c2[bcd]a×a)=0
00=0
it is parallel to a

Let's check given expression is parallel to b
b×(2[abd]c2[bcd]a)=0
Here [abd] and [bcd] are scalar
So, b×(2pc2ra)=0
2pb×c2rb×a0
It is not parallel to b

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