Mathematics - Vector Question with Solution | TestHub

The equations of given lines in vector form may be written as $$L_1:\mathbf{r}(-\hat{\mathbf{i}}-2\hat{\mathbf{j}}-\hat{\mathbf{k}})+\lambda (3\hat{\mathbf{i}}+\hat{\mathbf{j}}+2\hat{\mathbf{k}})$$ and $L_2:\mathbf{r}=(2\hat{\mathbf{i}}-2\hat{\mathbf{j}}+3\hat{\mathbf{k}})+\mu (\hat{\mathbf{i}}+2\hat{\mathbf{j}}+3\hat{\mathbf{k}})$ $\therefore$ The vector perpendicular to both $L_1$ and $L_2$ is $$\left|\begin{array}{rrr}\hat{\mathbf{i}}& \hat{\mathbf{j}}& \hat{\mathbf{k}}\\ 3& 1& 2\\ 1& 2& 3\end{array}\right|=-\hat{\mathbf{i}}-7\hat{\mathbf{j}}+5\hat{\mathbf{k}}$$ $$\begin{array}{rl}\therefore \text{ Required unit vector }& =\frac{(-\hat{\mathbf{i}}-7\hat{\mathbf{j}}+5\hat{\mathbf{k}})}{\sqrt{(-1{)}^2+(-7{)}^2+(5{)}^2}}\\ & =\frac{1}{5\sqrt{3}}(-\hat{\mathbf{i}}-7\hat{\mathbf{j}}+5\hat{\mathbf{k}})\end{array}$$