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Question

If ¯a,¯b,¯c are non coplanar vectors, then find (¯¯¯a+2¯¯b¯¯c).[(¯¯¯a¯¯b)×(¯¯¯a¯¯b¯¯c)][¯¯¯a¯¯b¯¯c]

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Solution

Given
a,b,c are non coplanar vector
Hence, [abc]0
(a+2bc).[(ab)×(abc)][abc]
=(a+2bc).[(a×a)a×ba×cb×a+b×b+b×c][abc]
=(a+2bc).[0+b×a+c×ab×a+0+b×c][abc]
=(a+2bc).(b×c+c×a)[abc]
=a.(b×c)+a.(c×a)+2b.(b×c)+2b.(c×a)c.(b×c)c(c×a)[abc]
=[abc]+0+0+2[abc]00[abc]
=3

1067481_1181071_ans_3ae7ac756c404c7f952608223d2fd100.png

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