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2 problems match “scalar triple product”. Clear

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1. Dot product with a cross product of differences ●●○○○ 20 ptsLet $\mathbf{a} = \begin{pmatrix} -3 \\ 10 \\ 1 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 5 \\ \pi \\ 0 \end{pmatrix}$, and $\mathbf{c} = \begin{pmatrix} -2 \\ -2 \\ 7 \end{pmatrix}$. Compute $(\mathbf{a} - \mathbf{b}) \cdot [(\mathbf{b} - \mathbf{c}) \times (\mathbf{c} - \mathbf{a})]$.
2. Dot product with a cross product of unit vectors ●●●○○ 50 ptsLet $\mathbf{u}$ and $\mathbf{v}$ be unit vectors, and let $\mathbf{w}$ be a vector such that $\mathbf{u} \times \mathbf{v} + \mathbf{u} = \mathbf{w}$ and $\mathbf{w} \times \mathbf{u} = \mathbf{v}$. Compute $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$.