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Question

If a,b,c,d are four vectors, then prove that
(a×b).(c×d)+(b×c).(a×d)+(c×a).(b×d)=0

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Solution

The value of (axb).(cxd) can be found by
(axb).(cxd)= (a.c)(b.d)-(a.d)(b.c)
So the value for given problem will become
(a×b).(c×d)+(b×c).(a×d)+(c×a).(b×d)=
(a.c)(b.d)-(a.d)(b.c)+(b.a)(c.d)-(b.d)(c.a)+(c.b)(a.d)-(c.d)(a.b)
If we consider a.c=c.a= scalar (number)
Then all the above expression will mutually cancel and we will get the value as zero

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