Using a cross product to find another cross produc

Question

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.

Answer 1

Let me start if off for you:

$$\begin{align}(\vec{u} - 3\vec{v}) \times (\vec{u} + 2\vec{v}) &= (\vec{u} - 3\vec{v})\times u + (\vec{u} - 3\vec{v})\times(2\vec{v})\\&=(\vec{u} - 3\vec{v})\times \vec u + 2\cdot((\vec{u} - 3\vec{v})\times \vec v)\end{align}$$

In the first line, I used the second rule you wrote, and in the second, I used the first rule you wrote.

Now, continue doing that, and use two extra rules:

• $(\vec a + \vec b)\times \vec c = $ something
• $\vec a \times \vec a =$ something.