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Question

Solution of the vector equation x=x× a+b is

A
x=11+(a)2[(ab)a+b(a×b)]
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B
x=11+(b)2[(ab)b+a(a×b)]
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C
x=11+(b)2[(ab)b+a+(a×b)]
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D
x=11+(a)2[(ab)a+b+(a×b)]
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Solution

The correct option is A x=11+(a)2[(ab)a+b(a×b)]
Given eq
x=x×a+b---------(1)
cross product w.r. to a
x×a=(x×a)×a+b×a
x×a=(xa)a(aa)x+b×a
x×a=(xa)a(a)2x+b×a
from eq (1) putting x×a
xb+(a)2x=(xa)a(a×b)
x(1+(a)2)=((x×a+b)a)a+b(a×b)
x(1+(a)2)=((x×a)a+ba)a+b(a×b)
by triple scalar product (x×a)a=0
x(1+(a)2)=(ab)a+b(a×b)
x=(ab)a+b(a×b)(1+(a)2)
x=1(1+(a)2)[(ab)a+b(a×b)]


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