Question

a vector + b vector and a vector - b vector must lie in same plane . Comment ​

Answer 1

Answer:

What is the value of ( A + B ) . (A × B )

We know from scalar triple product, that for vectors A, B, and C,

A . ( B × C) = B . ( C × A) = C . ( A× B) = (A×B) . C

In our question

( A + B ) . ( A × B ) = A. ( A× B) + B . ( A × B)

Now

A. ( A× B) = B. ( A×A)= 0

and

B . ( A ×B ) = A. ( B× B)= 0

Therefore,

( A +B ) . ( A × B ) = 0 +0= 0

The dot product of the sum of two Vectors with the cross products of the same two vector is zero and a scalar.

We can understand the answer as below. The sum of two vectors A and B is a vector in the plane defined by A and B. Whereas A×B is a vector which is perpendicular to the plane of vectors A and B. The dot product of two vectors one in the plane of vectors A,B and the other perpendicular to the plane is zero. Furthermore being a dot product it is a scalar.