Do A+B andA-B lie in same plane? Explain.

Yes, and to prove this we can use a combination of the dot and cross product. 

 

So, if we assume that both A+B and A-B lie in the same plane, then

 

(A+B). ((A+B) x (A-B)) = 0

 

So, LHS = (A+B). ((AXA) - (AxB) + (B x A) - (BxB))

 

= (A+B). (0- (AxB) - (A x B) - 0) (Using commutative law of vector multiplication which says that, (axb) = -(bxa) and also a x a = 0)

 

= (A+B). (-2(AxB))

= -2 (A. (A x B) + B. (A x B))

=0  (Using that a. (a x b) = 0 as a x b gives a vector which is perpendicular to both a and b vectors and a dot product between perpendicular vectors is 0)

 

Hence proved.