I was reading triple cross product. I had read something just like this :
$$\vec A \times (\vec B \times \vec C)=\vec B(\vec A\cdot \vec C)-\vec C(\vec A\cdot \vec B)$$
which is also called "back of the cab". I know that cross product is vector, like as $$\vec C=\vec A\times \vec B$$ so C is vector. But when I used dot product they are forming a scalar. So my question is do we really use the expression (back of the cab) for magnitude of triple cross product?
$$C=\vec A\cdot \vec B$$
Editing again after seeing Jean's comment (first comment) :
When I used dot product that became scalar as I said earlier so I am going to assume
$$A\cdot B=\alpha \\ C\cdot A=\beta$$
alpha and beta is scalar now but A and B is scalar or vector, who cares? When I multiplied alpha and beta by B respectively with B then I got
$$B\cdot \alpha-\beta \cdot B$$ Now they are scalar. Who says they are vector?