Vectors and matrices › Lines, planes and vectors in space

Angles between vectors and planes

Solve the first problem 18 problems

These are 9 of the 18 problems. Sign in for free to see them all. Sign in

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. 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. Angle with the z-axis ●●○○○ 20 ptsA vector makes an angle of $\frac{\pi}{4}$ with each of the x-axis and the y-axis. Find the angle, in degrees, that the vector makes with the z-axis.
6. 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.
7. 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 $\begin{pmatrix} b \\ c \\ a \end{pmatrix}$.
8. 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.
9. 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.

More in Lines, planes and vectors in space

Coplanarity and triple productsCross products and unit vectorsDistance to a planeLines and planesLines and planes: true or false