Mathematics - Vector Question with Solution | TestHub

The equation of line passing through the point $\left(-3,2,3\right)$ and parallel to a line with direction ratios $3,3,-1$ will be,

$\frac{x+3}{3}=\frac{y-2}{3}=\frac{z-3}{-1}=\lambda$

Now let any point on the line will be, $M\left(3\lambda -3,3\lambda +2,3-\lambda \right)$

We know the direction ratios of the line joining the points $\left(x_1,y_1,z_1\right)$ and $\left(x_2,y_2,z_2\right)$ is given by

$\left(x_2-x_1,y_2-y_1,z_2-z_1\right)$

Therefore, the direction ratios of line PM,

D.R of $\mathrm{PM}\left(3\lambda -7,3\lambda -4,5-\lambda \right)$

Since, $\mathrm{PM}$ is perpendicular to the line

$\Rightarrow 3\left(3\lambda -7\right)+3\left(3\lambda -4\right)-1\left(5-\lambda \right)=0$

$\Rightarrow \lambda =2$

$\Rightarrow \mathrm{M}\left(3,8,1\right)$

Now distance of point P from point M,

$\Rightarrow \mathrm{PM}=\sqrt{{\left(3-4\right)}^2+{\left(8-6\right)}^2+{\left(1+2\right)}^2}$

$\Rightarrow \mathrm{PM}=\sqrt{14}$