Q.If Q is the foot of the perpendicular from P$(2,4,3)$ on the line joining the points A$(1,2,4)$ and B$(3,4,5)$, find coordinates of Q.

$P(2,4,3),A(1,2,4),B(3,4,5)$. Direction of $AB$: $(2,2,1)$. Let $Q$ divide $AB$ in ratio $t:1$ (from $A$):

$\overrightarrow{PQ}=Q-P=(2t-1,\ 2t-2,\ t+1)$. Perpendicularity to $AB$'s direction $(2,2,1)$:

$2(2t-1)+2(2t-2)+1(t+1)=0\Longrightarrow4t-2+4t-4+t+1=0\Longrightarrow9t-5=0\Longrightarrow t=\frac59.$ …

Parametrise the foot of the perpendicular as a point on line AB using an unknown ratio parameter t, form the vector from P to that point, set its dot product …

Setting up the perpendicularity condition using PQ dotted with PA or PB instead of …