Question

Find the equation of a line passing through the pointg $P(2,-1,3)$ and perpendicular to the lines $\overrightarrow{r}=\hat{i}+\hat{j}-\hat{k}+\lambda (2\hat{i}-2\hat{j}+\hat{k})$ and $\overrightarrow{r}=(2\hat{i}-\hat{j}-3\hat{k})+\mu (\hat{i}+2\hat{j}+2\hat{k})$

Answer 1

Let the equation of line passing through the point (2, -1, 3) be $\frac{x-2}{a}=\frac{y+1}{b}=\frac{z-3}{c}$ --- (1)

Given lines are

$\overrightarrow{r}=(\overrightarrow{i}+\overrightarrow{j}-\overrightarrow{k})+\lambda (2\overrightarrow{i}-2\overrightarrow{j}+\overrightarrow{k})$--- (2)

$\overrightarrow{r}=(2\overrightarrow{i}-\overrightarrow{j}-3\overrightarrow{k})+\mu (\overrightarrow{i}+2\overrightarrow{j}+2\overrightarrow{k})$--- (3)

Since (1), (2) and (3) are perpendicular to each other
$\Rightarrow 2a-2b+c=0$

$a+2b+2c=0$

$\Rightarrow \frac{a}{-4-2}=\frac{b}{1-4}=\frac{c}{4+2}$

$\Rightarrow \frac{a}{-6}=\frac{b}{-3}=\frac{c}{6}=l$

$\Rightarrow a=-6l,b=-3l,c=6l$

Putting it in (1) we get required equation of line as

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

$\Rightarrow \frac{x-2}{2}=y+1=\frac{z-3}{-2}$