Finding cross product with limited variables

Question

I'm trying to solve a problem involving cross product but I'm stuck on how to proceed.

The problem states, Assume $u$ x $w$ = $\begin{bmatrix}6\\-5\\7\end{bmatrix}$

then find:

$(5u - 3w)$ x $w =$ ?

I've been looking at properties of the cross product but still am confused as to how I proceed from here. Any help is appreciated, thank you!

Answer 1

Using the algebraic properties of the cross product we have:

$$ (5u-3w)\times w=(5u)\times w -(3w)\times w=5(u\times w)-3(w \times w) $$

and, since the self cross product is null: $$ (5u-3w)\times w=5(u\times w)=[30,-25,35]^T $$