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Q.

Let u^,v^,w^ be three non-coplanar unit vectors with angles between u^ and v^ is  α , between v^ and w^ is β and between w^ and u^ is γ . Further a→,b→,c→ be unit   vectors along angle bisectors of α,β,γ respectively. [a→×b→b→×c→c→×a→]=

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a

14[u^v^w^]2sec2⁡α2sec2⁡β2sec2⁡γ2

b

14[u^v^w^]2sec2⁡αsec2⁡βsec2⁡γ

c

116[u^v^w^]2sec2⁡α2sec2⁡β2sec2⁡γ2

d

14[w^u^v^]2sec⁡α2sec⁡β2sec⁡γ2

answer is C.

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Detailed Solution

[a→×b→b→×c¯c→×a→]=[a→b→c¯]2=((u^+v^)×(v^+w^))·(w^+u^)8cos⁡α2cos⁡β2cos⁡γ22=2[u^v^w^]8∣cos⁡α2cos⁡β2cos⁡γ2∣2=[u^v^w^]216sec2⁡(α/2)sec2⁡(β/2)sec2⁡(γ/2)
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