How to find the union&intersection of two lines by their equations? I will try to be as clear as possible concerning my confusion, and I will use some examples(several ones).
Case number 1.
Assume two equations(in cartesian form) of two planes.
$2x+2y-5z+2=0$ and $x-y+z=0$
Now,we need to find their vectors.
For the first on, we get: {1,-1,0}, {0,5/2,1} and {1,0,2/5}.
For the second equation, we get: {1,1,0}, {0,1,1},and {1,0,-1}.
Now, I have a hard time understanding how I have to figure out where is their intersection and their union?
Case 2:
Assume the one of the previous planes $x-y+z=0$ and the line  $x-y=0$. How do I find the intersection and union of these two?

Case 2 is simple:
$x-y+z=0$ and $x−y=0$ 
means that (substitute second into first) 
$z=0$, 
i.e. your solution lies on the plane $z=0$. This means that your solution is just the line $y=x$ since $x−y=0$.
You can see this because $x-y+z=0$ is a plane that intersects the $z=0$ plane exactly on that line. 
Since the line defined by $x−y=0$ is contained in the plane $x-y+z=0$, then the union is just the plane $x-y+z=0$.
