Question

Find the equation of the line through the point (1,-1,1) and perpendicular to the lines joining the points (4,3,2),(1,-1,0) and (1,2,-1),(2,1,1).

Answer 1

The d.r's of line $L_1$ joining (4,3,2) and (1,-1,0) are 3,4,2
The d.r's of line $L_2$ joining (1,2,-1) and (2,1,1) are 1,-1,2.
A vector perpendicular to both the lines is $\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 3& 4& 2\\ 1& -1& 2\end{array}\right|=10\hat{i}-4\hat{j}-7\hat{k}$

$\therefore$eq.of line through (1,-1,1) and $\perp L_1\text{and}{L}_{2}is:\overrightarrow{r}=(\hat{i}-\hat{j}+\hat{k})+\lambda (10\hat{i}-4\hat{j}-7\hat{k})$