How to evaluate $(\mathbf{a}\times\mathbf{b})\cdot (\mathbf{a} - \mathbf{b})$

Question

The answer is $\mathbf 0$ but I don't understand how to get to this answer.

If the answer is $\mathbf 0$ then that means $(\mathbf{a}-\mathbf{b})$ is perpendicular to $(\mathbf{a}\times\mathbf{b})$, can someone please explain to me why $(\mathbf{a}-\mathbf{b})$ is perpendicular to $(\mathbf{a}\times\mathbf{b})$

Answer 1

In fact there is an easy reason why this must be true: $a-b$ lies in the plane of $a,b$, while $a \times b$ is perpendicular to that whole plane.

Answer 2

$$(a\times b).(a-b)$$is a determinant with rows being $a,b,a-b$

That determinant is zero because the rows are linearly dependent.

Answer 3

When two vectors are added they lie on the same plane which the original two vectors are placed.(Therefore (a-b),(a+b),(2a+b).... all lie on the same plane which a and b are on.)

Further, the cross product of two vectors is perpendicular to both the original vectors.(a cross b results a vector and that vector is perpendicular to both a and b.)

Then we know that dot product of two parallel vectors is zero.(Dot product results a scalar.)

By the above facts we can derive your answer.