Angle Between Two Lines

Find the angle between the following pair of lines

$\overrightarrow{r}=\left(2+s\right)\hat{i}+\left(s-1\right)\hat{j}+\left(2-3s\right)\hat{k}$

$\overrightarrow{r}=\left(1-t\right)\hat{ı}+\left(2t+3\right)\hat{ȷ}+\hat{k}$

Find the angle between the pair of lines $\frac{x-2}{2}=\frac{y-1}{5}=\frac{z+3}{-3}$ and $\frac{x+2}{-1}=\frac{y-4}{8}=\frac{z-5}{4}$.

Find the angle between the pair of lines $\frac{x}{2}=\frac{y}{2}=\frac{z}{1}$ and $\frac{x-5}{4}=\frac{y-2}{1}=\frac{z-3}{8}$.

Find the angle between the following pair of lines $\overrightarrow{r}=2\hat{ı}-5\hat{ȷ}+\hat{k}+\lambda \left(3\hat{ı}+2\hat{ȷ}+6\hat{k}\right)$ and $\overrightarrow{r}=7\hat{i}-6\hat{k}+\mu \left(\hat{i}+2\hat{ȷ}+2\hat{k}\right)$.

Find the angle between the following pair of lines $\overrightarrow{r}=3\hat{i}+\hat{ȷ}-2\hat{k}+\lambda \left(\hat{i}-\hat{ȷ}-2\hat{k}\right)$ and $\overrightarrow{r}=2\hat{ı}-\hat{ȷ}-56\hat{k}+\mu \left(3\hat{i}-5\hat{ȷ}-4\hat{k}\right)$.

Find the angle between the lines whose direction ratios are $\left(a,b,c\right)$ and $\left(b-c,c-a,a-b\right)$.

If the coordinates of the points $A, B, C, D$ be $\left(1,2,3\right), \left(4,5,7\right), \left(-4,3,-6\right)$ and $\left(2,9,2\right)$ respectively, then find the angle between $AB$ and $CD$.

If the lines $\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}$ and $\frac{x-1}{3k}=\frac{y-1}{1}=\frac{z-6}{-5}$ are perpendicular, find the value of $k$.

Find the angle between the lines $\overrightarrow{r}=4\hat{i}-\hat{ȷ}+\lambda \left(\hat{i}+2\hat{j}-2\hat{k}\right)$ and $\overrightarrow{r}=\hat{i}-\hat{ȷ}+2\hat{k}-\mu \left(2\hat{ı}+4\hat{ȷ}-4\hat{k}\right)$.

Find the angle between the lines

$\frac{x-3}{1}=\frac{y-2}{2}=\frac{z-2}{-4}$ and $\frac{x-0}{3}=\frac{y-5}{2}=\frac{z+2}{-6}$.

Find the angle between the lines

$\frac{x+4}{3}=\frac{y-1}{5}=\frac{z+3}{4}$ and $\frac{x+1}{1}=\frac{y-4}{1}=\frac{z-5}{2}$.

Find the angle between the following pair of lines

$\frac{x+1}{3}=\frac{y-1}{5}=\frac{z+3}{4} ; \frac{x+1}{1}=\frac{y-4}{4}=\frac{z-5}{2}$.

Find the angle between the following pair of lines:

$\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-4}{6} ; x+1=\frac{y+2}{2}=\frac{z-4}{2}$

Find the angle between the following pair of lines:

$\overrightarrow{r}=5\hat{i}-7\hat{j}+\lambda \left(-\hat{i}+4\hat{j}+2\hat{k}\right)$

$\overrightarrow{r}=2\hat{ı}+\hat{k}+\mu \left(3\hat{ı}+4\hat{k}\right)$

Show that the line joining the origin to the point $(2,1,1)$ is perpendicular to the line determined by the points $(3,5,-1)$ and $(4,3,-1)$.

Show that the line $\frac{x-3}{2}=\frac{y+1}{-3}=\frac{z-2}{4}$ is perpendicular to the line $\frac{x+2}{2}=\frac{y-4}{4}=\frac{z+5}{2}$.

The acute angle between the lines $\overrightarrow{r}=(4\hat{i}-\hat{j})+5(2\hat{i}+\hat{j}-3\hat{k})$ and $\overrightarrow{r}=(\hat{i}-\hat{j}+2\hat{k})+t(\hat{i}-3\hat{j}+2\hat{k})$ is

The line $\frac{x-x_1}{0}=\frac{y-y_1}{1}=\frac{z-z_1}{2}$ is

The line $x=1$, $y=2$ is

The angle between the lines $x=1$, $y=2$ and $y+1=0$, $z=0$ is