Proving vectors $a$ and $b(a \cdot c) – c(a \cdot b)$ are perpendicular

Question

I am trying to prove, but can't seem to do it.

Prove vectors $a$ and $b(a \cdot c) – c(a \cdot b)$ are perpendicular.

This is where I am stuck.

$$|\vec{a}|*|\vec{b}(|\vec{a}||\vec{c}|cos(\angle{ab})|-\vec{c}(|\vec{a}||\vec{b}|cos(\angle{ab}))|*cos(\angle(b(a \cdot c) – c(a \cdot b) )=0$$

Answer 1

We have that

$$a \cdot [b(a · c) – c(a · b)]=(a\cdot b)(a · c) – (a\cdot c)(a · b)=0$$

therefore the vectors are orthogonal.

Answer 2

Note that$$a.\bigl(b(a.c)-c(a.b)\bigr)=(a.b)(a.c)-(a.c)(a.b)=0.$$