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Question

A(a), B(b), C(c) are the vertices of a triangle ABC and R(r) is any point in the plane of triangle ABC, then r.(a×b+b×c+c×a) is always equal to


A

Zero

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B

[a b c]

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C

[ a b c]

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D

None of these

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Solution

The correct option is B

[a b c]


A vector perpendicular to the plane of A(a), B(b), C(c) is (ba)×(ca)=a×b+b×c+c×a
Now for any point R(r) in the plane of A, B, C
(ra).(a×b+b×c+c×a)=0r.(a×b+b×c+c×a)a.(a×b+b×c+c×a)=0r.(a×b+b×c+c×a)=0+a.b×c+0r.(a×b+b×c+c×a)=[a b c]


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