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Question

If a and b are perpendicular unit vectors and vector c is such that c=a+b, then (a×b).(b×c)+(b×c).(c×a)+(c×a).(a×b) is

A
0
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B
1
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C
-1
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D
b.c+c.a
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Solution

The correct option is C -1
(a×b).(b×c)+(b×c).(c×a)+(c×a).(a×b)
=(a.b)(b.c)(a.c)(b.b)+(b.c)(c.a)(b.a)(c.c)+(c.a)(a.b)(c.a)(a.b)(a.a)
=0(a.c)+(b.c)(c.a)0+0(c.b)(a.b=0)
=(b.c)(c.a)(c.a)(b.c)+11
=((b.c)1)((c.a)1)1
=(11)(11)1=1(c=a+b)

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