### 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: To find $(\mathbf{v} + \mathbf{w}) \times (\mathbf{v} + \mathbf{w}),$ I need to use the properties of the cross product.
I recall that the cross product of two parallel vectors is zero, so $(\mathbf{v} + \mathbf{w}) \times (\mathbf{v} + \mathbf{w}) = \mathbf{0}.$
Alternatively, I could use the distributive property and the anticommutative property to expand the expression as follows: $(\mathbf{v} + \mathbf{w}) \times (\mathbf{v} + \mathbf{w}) = \mathbf{v} \times \mathbf{v} + \mathbf{v} \times \mathbf{w} + \mathbf{w} \times \mathbf{v} + \mathbf{w} \times \mathbf{w} = \mathbf{0} + \mathbf{v} \times \mathbf{w} - \mathbf{v} \times \mathbf{w} + \mathbf{0} = \mathbf{0}.$