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Question:** A cloud computing system routes data along vectors in $\mathbb{R}^3$. Given $\mathbf{v} \times \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix} = \begin{pmatrix} 5 \\ -6 \\ -1 \end{pmatrix}$, find $\mathbf{v}$.
Solution:** Let $\mathbf{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ and $\mathbf{a} = \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}$. Then:
\mathbf{v} \times \mathbf{a} = \begin{pmatrix} y(4) - z(-1) \\ z(2) - x(4) \\ x(-1) - y(2) \end{pmatrix} = \begin{pmatrix} 4y + z \\ 2z - 4x \\ -x - 2y \end{pmatrix} = \begin{pmatrix} 5 \\ -6 \\ -1 \end{pmatrix}

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