The foot of the perpendicular drawn from a point with position vector $\overline{i}+4\overline{k}$ on the line joining the points  $\overline{j}+3\overline{k};2\overline{i}-3\overline{j}+\overline{k}$ is 

A (1, 0, 4), B (0, 1, 3), C(2, -3, 1)

Dr’s of $\overline{BC}=\left(2,-4,-2\right)$

$=\left(-1,2,1\right)$

Equation of  $\overline{BC}=\frac{x}{-1}=\frac{y-1}{2}=\frac{z-3}{1}$

$P=\left(-t,2t+1,t+3\right)$

Dr’s of  $AP=\left(-t-1,2t+1,t-1\right)$

$\overline{AP}\perp \overline{BC}$

$-\left(-t-1\right)+2\left(2t+1\right)+1\left(t-1\right)=0$

$t+1+4t+2+t-1=0$

$6t+2=0$

$t=-\frac{1}{3}$

$P=\left(\frac{1}{3},\frac{1}{3},\frac{8}{3}\right)$

i.e $\frac{1}{3}\left(i+j+8k\right)$