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1. Area of a parallelogram from vectors ●○○○○ 10 ptsCalculate the area of the parallelogram spanned by the vectors $\vec{a} = (3, -3, 1)$ and $\vec{b} = (4, 9, 2)$. Round to two decimal places.
2. Cross product of two vectors ●○○○○ 10 ptsFind $(2\hat{i}+3\hat{j}-\hat{k})\times(3\hat{j}+2\hat{k})$. Write the result as $a\hat{i}+b\hat{j}+c\hat{k}$ and enter the coefficients $a$, $b$, $c$.
3. Area of a space quadrilateral ●●○○○ 20 ptsThe points $A = (3,-4,2)$, $B = (5,-8,5)$, $C = (4,-3,0)$, and $D = (6,-7,3)$ in space form a flat quadrilateral. Find the area of this quadrilateral. Round to two decimal places.
4. Dot product from norms and cross product ●●○○○ 20 ptsLet $\mathbf{a}$ and $\mathbf{b}$ be vectors such that $\|\mathbf{a}\| = 2,$ $\|\mathbf{b}\| = 5,$ and $\|\mathbf{a} \times \mathbf{b}\| = 8.$ Find $|\mathbf{a} \cdot \mathbf{b}|.$
5. 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 $(\mathbf{a} - \mathbf{b}) \cdot [(\mathbf{b} - \mathbf{c}) \times (\mathbf{c} - \mathbf{a})]$.
6. Area of parallelogram from its diagonals ●●○○○ 20 ptsLet $\mathbf{p}$ and $\mathbf{q}$ be two three-dimensional unit vectors such that the angle between them is $30^\circ.$ Find the area of the parallelogram whose diagonals correspond to $\mathbf{p} + 2 \mathbf{q}$ and $2 \mathbf{p} + \mathbf{q}.$ Give your answer as a decimal.
7. Area of a parallelogram from vector combinations ●●○○○ 20 ptsCalculate the area of the parallelogram constructed on the vectors $\vec{a}=3\vec{p}+2\vec{q}$ and $\vec{b}=2\vec{p}-\vec{q}$, given that $|\vec{p}|=4$, $|\vec{q}|=3$ and the angle between $\vec{p}$ and $\vec{q}$ is $\frac{3\pi}{4}$. Round to two decimal places.
8. 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 $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$.
9. 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.