Angle between Two Lines

Find the acute angle between the line joining points $\left(2, 1, 3\right)$ and $\left(1, -1, 2\right)$ and the line having direction ratios $2, 1, -1$.         [Enter the value in degrees excluding degree symbol]

Find the angle between the pair of lines $\overline{r}=3\hat{i}+2\hat{j}-4\hat{k}+\lambda (\hat{i}+2\hat{j}+2\hat{k})$ and $\overline{r}=5\hat{i}-2\hat{k}+\mu (3\hat{i}+2\hat{j}+6\hat{k})$.
 

If $l_1, m_1, n_1;l_2, m_2, n_2$ and $l_3, m_3, n_3$ are direction cosines of three mutually perpendicular lines $OA, OB, OC$, show that the line $OP$ whose direction cosines are proportional to $l_1+l_2+l_3, m_1+m_2+m_3, n_1+n_2+n_3$ makes equal angles with lines $OA, OB$ and $OC$.

Directions ratios of two lines satisfy the relation $2a-b+2c=0$ and $ab+bc+ca=0$. Show that the lines are perpendicular.

Find the direction cosines of the line which is perpendicular to the lines with direction ratios $4, 1, 3$ and $2, -3, 1$.

Find the angle between the lines whose direction ratios are $4, -3, 5$ and $3, 4, 5$.         [Enter the value in degrees excluding degree symbol]

If the direction ratios of two vectors are connected by the relations $p+q+r=0$ and $p^2+q^2-r^2=0,$ find the angle between them.

Find the direction cosines of the vector which is perpendicular to the vectors with direction ratios $-1, 2, 2$ and $0, 2, 1$.

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are $1, 3, 2$ and $-1, 1, 2$.

If a line drawn from the point $A(1, 2, 1)$ is perpendicular to the line joining $P(1, 4, 6)$ and $Q(5, 4, 4),$ then find the coordinates of the foot of the perpendicular.

Find $x$, if $\Delta ABC$ is right angled at $A,$ where $A\equiv (4, 2, 3), B\equiv (3, 1, 8), C\equiv (x, -1, 2)$.

Find $k,$ if $∆ABC$ is right angled at $B,$ where $A\equiv (5, 6, 4), B\equiv (4, 4, 1), C\equiv (8, 2, k)$.

Show that the vector $\overline{AB}$ is perpendicular to $\overline{CD}$ where $A\equiv (3, 4, -2), B\equiv (1, -1, 2), C\equiv (0, 3, 2)$ and $D\equiv (3, 5, 6)$.

If the angle between the vectors $\overline{a}$ and $\overline{b}$ having direction ratios $1, 2, 1$ and $1,3k, 1$ is $\frac{\pi }{4},$ find $k$.

Find the measure of acute angle between the lines whose direction ratios are $1, 2, 2$ and $-3, 6, -2$.

Find the measure of acute angle between the lines whose direction ratios are $3, 2, 6$ and $-2, 1, 2$. 

If the angle between the lines whose direction cosines are given by the equation $l+m+n=0,l^2+m^2-n^2=0$ is $\frac{\pi }{k}$, then find the value of $k$.

If the angles between the lines $\overline{r}=3\hat{i}-2\hat{j}+6\hat{k}+\lambda (2\hat{i}+\hat{j}+2\hat{k})$ and $\overline{r}=(2\hat{j}-5\hat{k})+\mu (6\hat{i}+3\hat{j}+2\hat{k})$ is $\theta ={\cos }^{-1}k$, then the value of $k$ is

If the angle between the lines $\overrightarrow{r}=3\hat{i}-2\hat{j}+6\hat{k}+\lambda (2\hat{i}+\hat{j}+2\hat{k})$ and $\overrightarrow{r}=(2\hat{j}-5\hat{k})+\mu (6\hat{i}+3\hat{j}+2\hat{k})$ is $\theta ={\cos }^{-1}k$, then the value of $k$ is

Find the value of $7k$ so that the lines are at right angles.

$\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}$ and $\frac{x-1}{3k}=\frac{y-1}{1}=\frac{z-6}{-5}$