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2 problems match “vector triple product”. Clear

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1. 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 $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$.
2. Sine of angle from triple product ●●●○○ 50 ptsLet $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be nonzero vectors, no two of which are parallel, such that $$(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \frac{1}{3} \|\mathbf{b}\| \|\mathbf{c}\| \mathbf{a}.$$ Let $\theta$ be the angle between $\mathbf{b}$ and $\mathbf{c}.$ Find $\sin \theta.$ Round to two decimal places.