Angle between Two Lines

If the straight line joining the points $\left(2,1,4\right) \mathrm{and} \left(a-1,4,-1\right)$ is parallel to the line joining the points $\left(0,2,b-1\right)$ and $\left(5,3,-2\right)$, find the values of $a+3b$

Find the acute angle between the following lines.

$\overrightarrow{r}=\left(\hat{i}-\hat{j}\right)+t\left(-2\hat{i}+2\hat{j}+\hat{k}\right),$

$\overrightarrow{r}=\left(\hat{i}-2\hat{j}+\hat{k}\right)+s\left(2\hat{i}-2\hat{j}-\hat{k}\right)$.

The acute angle between the lines $\overrightarrow{r}=\left(4\hat{i}-\hat{j}\right)+t\left(\hat{i}+2\hat{j}-2\hat{k}\right)$ and$\overrightarrow{r}=\left(\hat{i}-2\hat{j}+4\hat{k}\right)+s\left(-\hat{i}-2\hat{j}+2\hat{k}\right)$ is

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

Find the value of $\lambda ,$ so that the lines $\frac{1-x}{3}=\frac{7y-14}{2\lambda }=\frac{z-3}{2}$ and $\frac{7-7x}{3\lambda }=\frac{y-5}{1}=\frac{6-z}{5}$ are at right angle.

The angle between the lines $\frac{x-1}{4}=\frac{y-3}{1}=\frac{z}{8}$ and $\frac{x-2}{2}=\frac{y+1}{2}=\frac{z-4}{1}$ is

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

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 $A\equiv (5, 6, 4), B\equiv (4, 4, 1), C\equiv (8, 2, k)$.