Question

Show that line $\overrightarrow{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda (3\hat{i}-\hat{j})$ and $\overrightarrow{r}=(4\hat{i}-\hat{k})+\mu (2\hat{i}+3\hat{k})$ intersect. Also find their point of intersection.

Answer 1

$\frac{x-1}{3}=\frac{y-1}{-1}=\frac{z+1}{0}=\lambda$ ---- (1)

$\frac{x-4}{2}=\frac{y-0}{0}=\frac{z+1}{3}=\mu$ --- (2)

Point on first line,

$[3\lambda +1,-\lambda +1,-1]$---- (3)

Point on second line,

$[2\mu +4,0,3\mu -1]$---- (4)

For point of intersection, equation $3=$ equation $4.$

We get, $\lambda =1,\mu =0$

Therefore, point of intersection is $(4,0,-1).$