Let $\mathbf{a} = \begin{pmatrix} 2 \\ 1 \\ 5 \end{pmatrix}.$  Find the vector $\mathbf{b}$ such that $\mathbf{a} \cdot \mathbf{b} = 11$ and
\[\mathbf{a} \times \mathbf{b} = \begin{pmatrix} -13 \\ -9 \\ 7 \end{pmatrix}.\]
Question:
Let $\mathbf{a} = \begin{pmatrix} 2 \\ 1 \\ 5 \end{pmatrix}.$  Find the vector $\mathbf{b}$ such that $\mathbf{a} \cdot \mathbf{b} = 11$ and
\[\mathbf{a} \times \mathbf{b} = \begin{pmatrix} -13 \\ -9 \\ 7 \end{pmatrix}.\]
Answer:
Let's think step by step.

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\begin{pmatrix} -1 \\ 3 \\ 2 \end{pmatrix}
Let $\mathbf{a} = \begin{pmatrix} 2 \\ 1 \\ 5 \end{pmatrix}.$  Find the vector $\mathbf{b}$ such that $\mathbf{a} \cdot \mathbf{b} = 11$ and
\[\mathbf{a} \times \mathbf{b} = \begin{pmatrix} -13 \\ -9 \\ 7 \end{pmatrix}.\]
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