Equation of a plane as matrix.

What matrix P projects every point in $R^3$ onto the line of intersection of the planes $x+y+t =0$ and $x−t =0$?

I know how to solve it, but I'm not sure whether or not I correctly understand the meaning of plane equation. Isn't $x+y+t=0$ the same as undetermined system $Ax=0$, where $1$-by-$3$ matrix $A$ of rank $1$ have infinitely many solutions(planes) for its null space? Also, as we construct plane as a subspace of $R^3$ it should be done with two independent vectors, does this applie for planes which are not going through the origin?

• 1. Yes, there are many solutions: each solution is a vector that belongs to the plane. 2. Planes that do not go through the origin belong to affine geometry, their definition requires two vectors and a point that plays the role of the origin... but the one in your problem does go through the origin. – Miguel May 14 '17 at 9:21
• Hello Miguel, thank you for response and clarification, what if i substitute $x=1,y=1,t=-2$ or $x=-2,y=1,t=1$, isn't this different planes? No need to answering this i just understand everything :) Thank you – Anatoly Strashkevich May 14 '17 at 9:29
• How do you project a point onto a line? Do you mean find the closest point on the line to the point, or is there an orthogonal projection direction specified? – ja72 May 15 '17 at 20:31