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1. Angle between two lines in space ●○○○○ 10 ptsFind the angle between the lines $L_1: x=-1+3t,\ y=1+4t,\ z=-2t$ and $L_2: x=-1+2s,\ y=3s,\ z=-7+s$, taken as the angle between their direction vectors $\langle 3,4,-2\rangle$ and $\langle 2,3,1\rangle$. Give the angle in degrees. Round to one decimal place.
2. Angle between two vectors ●○○○○ 10 ptsGiven vectors $\overrightarrow{a} = (0, 2, 1)$ and $\overrightarrow{b} = (-1, 1, -2)$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ in degrees.
3. Dot product with a linear combination ●○○○○ 10 ptsGiven that the angle between vectors $\vec{a}$ and $\vec{b}$ is $120^{\circ}$, and $|\vec{a}|=2$, $|\vec{b}|=5$, find $(2\vec{a}-\vec{b})\cdot\vec{a}$.
4. Angle between a unit vector and a given vector ●○○○○ 10 ptsLet $\overrightarrow{a}$ be a unit vector and $\overrightarrow{b}=(1, \sqrt{3})$. Given that $\overrightarrow{a} \cdot \overrightarrow{b}=1$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ in degrees.
5. Dot product using a dual basis ●○○○○ 10 ptsLet $B=\{\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3,\mathbf{e}_4\}$ be a basis of $\mathbb{R}^4$ with $\mathbf{e}_1=(1,0,0,0)$, $\mathbf{e}_2=(1,1,0,0)$, $\mathbf{e}_3=(1,1,1,0)$, $\mathbf{e}_4=(1,1,1,1)$. Its dual basis is $\mathbf{e}^1=(1,-1,0,0)$, $\mathbf{e}^2=(0,1,-1,0)$, $\mathbf{e}^3=(0,0,1,-1)$, $\mathbf{e}^4=(0,0,0,1)$. Let $\mathbf{u}=\mathbf{e}^1-\mathbf{e}^2+\mathbf{e}^3-\mathbf{e}^4$ and $\mathbf{v}=(1,2,3,4)$, where vectors are written in the standard basis of $\mathbb{R}^4$. Find $\mathbf{u}\cdot\mathbf{v}$ using the standard dot product.
6. Cosine of angle between vectors ●●○○○ 20 ptsFor certain vectors $\mathbf{p}$ and $\mathbf{q}$, the vectors $3 \mathbf{p} + \mathbf{q}$ and $5 \mathbf{p} - 3 \mathbf{q}$ are orthogonal. Also, the vectors $2 \mathbf{p} + \mathbf{q}$ and $4 \mathbf{p} - 2 \mathbf{q}$ are orthogonal. If $\theta$ is the angle between $\mathbf{p}$ and $\mathbf{q}$, find $\cos \theta$. Give your answer as a decimal.
7. Cosine of angle with orthogonal vector sum ●●○○○ 20 ptsLet $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be three vectors with equal magnitudes, such that any two of them are orthogonal. If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{a} + \mathbf{b} + \mathbf{c},$ find $\cos \theta.$ Round to two decimal places.
8. Triangle area from a dot product ●●○○○ 20 ptsIn triangle $ABC$, $AB=1$, $BC=3$, and $\overrightarrow{AB}\cdot\overrightarrow{BC}=-1$. Find the area of triangle $ABC$. Round to two decimal places.
9. 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 $|\mathbf{a} \cdot \mathbf{b}|.$
10. 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 $(\mathbf{a} - \mathbf{b}) \cdot [(\mathbf{b} - \mathbf{c}) \times (\mathbf{c} - \mathbf{a})]$.

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