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1. Area of a triangle from vectors ●○○○○ 10 ptsLet $\mathbf{a} = \begin{pmatrix} 5 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}$. Find the area of the triangle with vertices $\mathbf{0}$, $\mathbf{a}$, and $\mathbf{b}$.
2. Weighted inner product of a sum ●○○○○ 10 ptsLet $\langle u,v\rangle = 4u_1v_1 + 5u_2v_2$ for vectors $u=(u_1,u_2)$ and $v=(v_1,v_2)$ in $\mathbb{R}^2$. Given $u=(3,-2)$, $v=(4,5)$ and $w=(-1,6)$, find $\langle u+v, w\rangle$.
3. 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.
4. 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.
5. 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 $(2\vec{a}-\vec{b})\cdot\vec{a}$.
6. 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.
7. 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.
8. Inclination angle from a direction vector ●○○○○ 10 ptsThe direction vector of the line $l$ has coordinates $(2, 2\sqrt{3})$. Find the inclination angle of the line $l$ in degrees, where the angle lies in $[0^\circ, 180^\circ)$.
9. 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.
10. 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 $a$, $b$, $c$.

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