Given $U \times V=\langle -4,-2,-1 \rangle$, find $(U-3V) \times (U+2V)$

Question

I am studying cross products and ran into trouble with this question. $U$ and $V$ are vectors, the $\times$ stands for the cross product. Thank you for your help in advance.

Edit: I assume I need to use the properties of the cross product in some way but I am not sure which ones to use. Specifically, I am confused as to how I am supposed to solve the interior equations of $(U-3v)$ and $(U+2V)$ without ever having been giving their original vectors. To my knowledge, it's not really possible to reverse the cross product if you are not given (at least 1 of) the original vectors used.

Answer 1

We can just use the facts that,

$V×V=0$, $U×U=0$ and $U×V=-V×U$

So,

$(U-3V)×(U+2V)$

$=U×(U+2V)-3V×(U+2V)$

$=U×U+2(U×V)-3(V×U)-6(V×V)$

$=2\langle -4,-2,-1\rangle+3\langle -4,-2,-1\rangle$

$=\langle -20,-10,-5\rangle$