Question

The perpendicular distance of the point whose position vector is $(1,3,5)$ from the line $\vec{r}=\hat{i}+2 \hat{j}+3 \hat{k}+\lambda$ $(\hat{i}+2 \hat{j}+2 \hat{k})$ is equal to

Answer 1

$\therefore P M=\left|\vec{v}_{2}\right| \sin \theta=\sqrt{5} \sin \theta$
As, $\cos \theta=\frac{\vec{v}_{1} \cdot \vec{v}_{2}}{\left|\vec{v}_{1}\right|\left|\vec{v}_{2}\right|}=\frac{6}{3 \sqrt{5}}=\frac{2}{\sqrt{5}}$
$\Rightarrow \sin \theta=\frac{1}{\sqrt{5}}$

$\therefore P M=\left|\vec{v}_{2}\right| \sin \theta=\sqrt{5}\left(\frac{1}{\sqrt{5}}\right)=1$