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Question

If a, b, c are non-coplanar vectors and ^d is a unit vector, then the value of (a.^d)(b×c)+(b.^d)(c×a)+(c.^d)(a×b) is equal to

A
1
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B
1
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C
[abc]
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D
none of these
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Solution

The correct option is C [abc]
Since a,b,c are non-coplanar, a×b,b×c,c×a are also non-coplanar, d can be expressed uniquely in terms of a,b and c.
Suppose d=x(b×c)+y(c×a)+x(a×b)
Then a.d=x[abc],b.d=y[abc],c.d=z[abc]
|(a.d)(b×c)+(b.d)(c×a)+(c.d)(a×b)|=|[abc][x(b×c)+y(c×a)+x(a×b)]|=|[abc]d|=|[abc]|×1=|[abc]|

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