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Question
If →a,→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:
If →a,→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:
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Solution
The correct option is A →a
Given (→a×→b)×(→c×→d)+(→a×→c)×(→d×→b)+(→a×→d)×(→b×→c)=(→a⋅(→c×→d))→b−(→b⋅(→c×→d))→a+(→b⋅(→a×→c))→d−(→d⋅(→a×→c))→b+(→a⋅(→b×→c))→d−(→d⋅(→b×→c))→a=[→a→c→d]→b−[→b→c→d]→a+[→b→a→c]→d−[→d→a→c]→b+[→a→b→c]→d−[→d→b→c]→a=−[→b→c→d]→a+[→a→b→c]→d+[→a→b→c]→d−[→b→c→d]→a=−2[→b→c→d]→a+2[→a→b→c]→don taking cross product with →a=−2[→b→c→d](→a×→a)+2[→a→b→c](→d×→a)Let vector →a is parallel to →dthen the cross product between parallel vector is zero=−2[→b→c→d](→a×→a)+2[→a→b→c](0)=0 Hence, the answer is →a.
Given
(→a×→b)×(→c×→d)+(→a×→c)×(→d×→b)+(→a×→d)×(→b×→c)
=(→a⋅(→c×→d))→b−(→b⋅(→c×→d))→a+(→b⋅(→a×→c))→d−(→d⋅(→a×→c))→b+(→a⋅(→b×→c))→d−(→d⋅(→b×→c))→a
=[→a→c→d]→b−[→b→c→d]→a+[→b→a→c]→d−[→d→a→c]→b+[→a→b→c]→d−[→d→b→c]→a
=−[→b→c→d]→a+[→a→b→c]→d+[→a→b→c]→d−[→b→c→d]→a
=−2[→b→c→d]→a+2[→a→b→c]→d
on taking cross product with →a
=−2[→b→c→d](→a×→a)+2[→a→b→c](→d×→a)
Let vector →a is parallel to →d
then the cross product between parallel vector is zero
=−2[→b→c→d](→a×→a)+2[→a→b→c](0)
=0
Hence, the answer is →a.
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