Q. If a line makes angles ${90}^{\circ },{135}^{\circ },{45}^{\circ }$ with the $x,y$ and $Z-\text{axes}$ respectively, find its direction cosines.

Q. Find the direction cosines of a line which makes equal angles with the coordinate axes.

Q. If a line has the direction ratios $-18,12,-4$, then what are its direction cosines?

Q. Find the values of $p$ so the line $\frac{1-x}{3}=\frac{7y-14}{2p}=\frac{z-3}{2}$ and $\frac{7-7x}{3p}=\frac{y-5}{1}=\frac{6-z}{5}$ are at right angles.

Q. Find the vector and the Cartesian equations of the lines that pass through the origin and $(5,-2,3)$

Q. Find the shortest distance between the lines whose vector equations are $\overrightarrow{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda (\hat{i}-3\hat{j}+2\hat{k})$ and $\overrightarrow{r}=(4\hat{i}+5\hat{j}+6\hat{k})+\mu (2\hat{i}+3\hat{j}+\hat{k})$.

Q. Find the Cartesian equation of the line which passes through the point $(-2,4,-5)$ and parallel to the line given by $\frac{x+3}{3}=\frac{y-4}{5}=\frac{z+8}{6}$

Q. Find the equation of the line in vector and in cartesian form that passes through the point with position vector $2\hat{i}-\hat{j}+4\hat{k}$ and is in the direction $\hat{i}+2\hat{j}-\hat{k}$.

Q. The cartesian equation of a line is $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$ write its vector form.

Q. Find the vector and the Cartesian equations of the line that passes through the points $(3,-2,-5),(3,-2,6)$

Q. Find the equation of the line which passes through the point $(1,2,3)$ and is parallel to the vector $(3\hat{i}+2\hat{j}-2\hat{k}$.

Q. $x+y+z=1$

Q. Find the shortest distance between the lines
$\overrightarrow{r}=(\hat{i}+2\hat{j}+\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k})$ and
$\overrightarrow{r}=\widehat{2i}-\hat{j}-\hat{k}+\mu (\widehat{2i}+\hat{j}+2\hat{k})$

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

Q. Find the shortest distance between the lines whose vector equations are :
$\overrightarrow{r}=(1-p)\hat{i}+(p-2)\hat{j}+(3-2p)\hat{k}$
$\overrightarrow{r}=(q+1)\hat{i}+(2q-1)\hat{j}-(2q-1)\hat{k}$

Q. Find the shortest distance between lines $\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}$ and $\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}$

Q. The angle between the following pair of lines:
$\frac{x}{2}=\frac{y}{2}=\frac{z}{1}$ and $\frac{x-5}{4}=\frac{y-2}{1}=\frac{z-3}{8}$ is $\theta ={\cos }^{-1}\frac{a}{3}$ then $a=$

Q. (i) $z=2$

Q. Find the angle between the following pairs of lines:
$\overrightarrow{r}=3\hat{i}+\hat{j}-2\hat{k}+\lambda (\hat{i}+\hat{j}-2\hat{k})$ and $\overrightarrow{r}=2\hat{i}-\hat{j}-56\hat{k}+\mu (3\hat{i}-5\hat{j}-4\hat{k})$.

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