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11 problems match “angle between vectors”. Clear

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1. 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.
2. 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 $(2\vec{a}-\vec{b})\cdot\vec{a}$.
3. 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.
4. 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.
5. 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.
6. 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 $|\mathbf{a} \cdot \mathbf{b}|.$
7. 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.
8. Angles of a unit vector with the axes ●●○○○ 20 ptsA unit vector $\vec a$ makes an angle $\frac{\pi}{3}$ with $\hat i$, an angle $\frac{\pi}{4}$ with $\hat j$, and an acute angle $\theta$ with $\hat k$. Find $\theta$ in degrees, and hence find the components of $\vec a$ along $\hat i$, $\hat j$, $\hat k$, in that order. Round the second component to two decimal places.
9. 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.
10. 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 $\begin{pmatrix} b \\ c \\ a \end{pmatrix}$.

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