How to find the the canonical equation of the projection of the line?

Question

How to find the the canonical equation of the projection of the line $$\frac{1}{3} \left(x - 4\right) = -\frac{1}{2} \left(y + 1\right) = \frac{z}{4}$$

on the plane $$x-3y-z+8=0.$$

Answer 1

Following the hint given by amd in the comment, two points on the line are given by

• $P=(4,-1,0)$
• $Q=(7,-3,4)$

to find the projections of the points onto the plane we can consider the lines through the two points and orthogonal to the plane (i.e. direction vector = normal vector for the plane)

• $P(t)=(4,-1,0)+t(1,-3,-1)$
• $Q(t)=(7,-3,4)+t(1,-3,-1)$

and then find by the intersections the projection $P_0$ and $Q_0$ onto the plane.