Q.Find the coordinates of the point where the line through the points A = (3,4,1) and B = (5,1,6) crosses the XY plane.

Parametrize the line through the two given points, then find the parameter value where $z=0$ (the defining equation of the $XY$-plane).

Step 1 — direction ratios. $A=(3,4,1),\ B=(5,1,6)$: direction ratios $=(5-3,\ 1-4,\ 6-1)=(2,-3,5)$.

Step 2 — parametric equations of the line.

$x=3+2t,\qquad y=4-3t,\qquad z=1+5t.$

Step 3 — set $z=0$ (the $XY$-plane).

$1+5t=0\ \Rightarrow\ t=-\frac15.$

Step 4 — find $x,y$ at this $t$. …