Line of intersection of two planes

[MarcL]

Hi,

I was doing a L.A question and a question arose. ( well I will write the question now, I found the answer I just can't visualize what I am doing which bothers me greatly)

Find the equation of the plane that contains the line (x,y,z)=(1,0,0)+t(1,3,2), and is parallel to the line of intersection of the two planes -x+2y+z=0 and x+z+1=0

Here is what bothers me: my partner found the answer by putting plane 1 and 2 in an augmented matrix and solving for it ( I'm guessing finding the parametric form). I just found the cross product of the two normals of the planes to find a line parallel to the plane that we are looking for.

How can 1) the augmented matrix find a line of intersection ( sorry if the question is broad I can't even understand the concept around that very well) and 2) How can a cross product define a line of intersection? Is there any proof to that? -- again sorry if the questions are hard ot understand / broad. Thank you for your help!

 

[homeomorphic]

The question is what does it mean for a vector to be in the plane? It's in the plane exactly when it's perpendicular to the normal vector. So if you have two planes, you have two normal vectors. When you take the cross product of the normal vectors, you get a vector that is perpendicular to both of them. So that means that it is in both planes, i.e. in the intersection of the planes. I think of it that way and just visualize the normal vectors, rather than trying to visualize the whole picture. But you can just think of a piece of paper with a needle sticking out of it to represent each plane and normal vector. It should then make sense that when you superimpose the two pieces of paper (one of them is a ghost piece of paper, so that it passes through the other), and put the needles together, the intersection is perpendicular to the two needles, so all you need to do is take the cross product of the needles.

As far as the augmented matrix, I think you may have to study more linear algebra to understand it better. Algebraically, all it is is that you have two equations, one for each plane, and the solutions for one of the equations are everything in that plane. So, when you solve BOTH equations, you get something that is in BOTH planes.

More conceptually, each equation is just saying that you have to be perpendicular to the normal vector in order to be in the plane. What you are looking for is things that are in both planes, or in other words, perpendicular to both normal vectors. This is where the linear algebra might get a little hairy for someone who hasn't studied it, but I'll attempt to explain it anyway. If you take all multiples and sums of the two normal vectors, you get a plane that will be perpendicular to the solution. The rows of the matrix are just the normal vectors, if we ignore the last column (which we don't really need, since it's all 0s and nothing interesting happens to it). If you do row operations, the rows will change to different vectors, but they will still span the same plane that's perpendicular to the solution. What happens is that using the row operations, which don't change the plane spanned by the row vectors, you can put it in a form where it's easy to see what's perpendicular to it.

 

[MarcL]

It actually all made sense. Thank you for taking the time to reply. I hate feeling like a robot and not understanding what I am doing. Feels kinda pointless. Especially that I would love to understand more what I am doing in Linear Algebra as I am going in software engineering next semester. Thank you!