### Instruction: If $\mathbf{v} \times \mathbf{w} = \begin{pmatrix} 5 \\ -2 \\ 4 \end{pmatrix},$ then find $(\mathbf{v} + \mathbf{w}) \times (\mathbf{v} + \mathbf{w}).$.### Response: Let's start by using the distributive property to expand the expression.
So we get $(\mathbf{v} + \mathbf{w}) \times (\mathbf{v} + \mathbf{w}) = \mathbf{v} \times \mathbf{v} + \mathbf{v} \times \mathbf{w} + \mathbf{w} \times \mathbf{v} + \mathbf{w} \times \mathbf{w}$.
Right. Now let's recall that the cross product is anti-commutative, so $\mathbf{v} \times \mathbf{w} = -\mathbf{w} \times \mathbf{v}$.
That means the middle two terms cancel and we are left with $\mathbf{v} \times \mathbf{v} + \mathbf{w} \times \mathbf{w}$.
Finally, recall that the cross product of a vector with itself is zero.