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11. Angle CBD in a tetrahedron ●●○○○ 20 ptsIn tetrahedron $ABCD$, $BC=1$, $BD=2$, $\angle ABC=90^{\circ}$, and $\overrightarrow{BC}\cdot\overrightarrow{DA}=-\sqrt{3}$. Find $\angle CBD$ in degrees.
12. 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.
13. Maximum angle between cyclic vectors ●●●○○ 50 ptsLet $a, b, c$ be nonzero real numbers. Find the maximum angle, in degrees, between the vectors $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$ and $\begin{pmatrix} b \\ c \\ a \end{pmatrix}$.
14. 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 $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$.
15. Dot product with circumcenter vector ●●●○○ 50 ptsIn triangle $ABC$, $AB = 3$ and $AC = 5$. Let $O$ be the circumcenter of triangle $ABC$. Find $\overrightarrow{OA} \cdot \overrightarrow{BC}$.
16. 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.
17. Side length from two dot products ●●●○○ 50 ptsIn $\triangle ABC$, $\overrightarrow{AB} \cdot \overrightarrow{AC} = \overrightarrow{AB} \cdot \overrightarrow{CB} = 4$. Find the length of side $AB$. Round to two decimal places.
18. Dot product in a regular tetrahedron ●●●○○ 50 ptsGiven a regular tetrahedron $P-ABC$ with edge length $1$, points $E, F, G, H$ lie on edges $PA, PB, CA, CB$ respectively. If $\overrightarrow{EF}+\overrightarrow{GH}=\overrightarrow{AB}$ and $\overrightarrow{EH} \cdot \overrightarrow{FG}=\frac{1}{18}$, find $\overrightarrow{EG} \cdot \overrightarrow{FH}$. Round to two decimal places.
19. Angle between lines from direction cosines ●●●○○ 50 ptsTwo lines have direction cosines $(l_1,m_1,n_1)$ and $(l_2,m_2,n_2)$. Each line satisfies $l+m+n=0$ and $\frac{mn}{q-r}+\frac{nl}{r-p}+\frac{lm}{p-q}=0$, where $p$, $q$, $r$ are distinct constants. Find the acute angle between the two lines, in degrees.
20. Coefficient from line-plane angle ●●●○○ 50 ptsThe angle between the plane $3x + by - 5z = 2$ and the straight line $x + 2 = \frac{3 - y}{2} = \frac{z - 1}{2}$ is $45^\circ$. Find the constant $b$, given that $0 < b < 10$.

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