# Using a cross product to find another cross produc

Given a cross product:

$$\vec{u} \times \vec{v} = \left< -1, 1, -3 \right>$$

I'm trying to find:

$$(\vec{u} - 3\vec{v}) \times (\vec{u} + 2\vec{v})$$

as a vector.

Clearly there are some properties of cross-products that I'm not aware of that would help solve this, but I can't for the life of me find them.

I do know the following rules:

$$(xa) \times b = x(a \times b) = a \times (xb)$$

and

$$a \times (b + c) = a \times b + a \times c$$

(where $$a$$, $$b$$, and $$c$$ are vectors and $$x$$ is a scalar)

But I have no idea how/if these rules apply to the above.

\begin{align}(\vec{u} - 3\vec{v}) \times (\vec{u} + 2\vec{v}) &= (\vec{u} - 3\vec{v})\times u + (\vec{u} - 3\vec{v})\times(2\vec{v})\\&=(\vec{u} - 3\vec{v})\times \vec u + 2\cdot((\vec{u} - 3\vec{v})\times \vec v)\end{align}
• $$(\vec a + \vec b)\times \vec c =$$ something
• $$\vec a \times \vec a =$$ something.