Question

Find the direction ratios of a line prependiclar to the two lines whose direction ratios are
(i) -2,1,-1 and -3,-4,1
(ii) 1, 3, 2 and -1, 1, 2

Answer 1

$\left(i\right)$ The given lines are $-2\hat{i}+\hat{j}-\hat{k}$ and $-3\hat{i}-4\hat{j}+\hat{k}$.

Now, we find the line perpendicular to the two lines :

$\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ -2& 1& -1\\ -3& -4& 1\end{array}\right|$ = $\hat{i}(1-4)-\hat{j}(-2-3)+\hat{k}(8+3)$

$=-3\hat{i}+5\hat{j}+11\hat{k}$

Hence, the direction ratios of the required line are $-3,5,11$.

$\left(ii\right)$ The given lines are $\hat{i}+3\hat{j}+2\hat{k}$ and $-\hat{i}+\hat{j}+2\hat{k}$.

Now, we find the line perpendicular to the two lines :

$\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 3& 2\\ -1& 1& 2\end{array}\right|$ = $\hat{i}(6-2)-\hat{j}(2+2)+\hat{k}(1+3)$

$=4\hat{i}-4\hat{j}+4\hat{k}$

Hence, the direction ratios of the required line are $4,-4,4$.