Understanding Plane Intersections in R3

[maccha]

thought I understood equations of planes in R3 and their intersections, but apparently not. I'm very confused by what seems to be a basic problem:

find a vector equation for the line of intersection of x + y + z= 0 and x + z = 0.

Is x + z= 0 still a plane even though it doesn't have the form Ax + By + Cz = D?

I notice that if you set the two equations equal to each other you find that y = 1. Does this mean that the planes intersect on a line where y =1 and all x coordinates are equal to negative z coordinates? Thanks!

 

[ystael]

The equation $$x + z = 0$$ does in fact have the form $$Ax + By + Cz = D$$; what is $$B$$ here?

Also, how did you combine $$x + y + z = 0$$ with $$x + z = 0$$ to conclude that $$y = 1$$? If you suppose that $$x + z = 0$$ and $$y = 1$$, what is $$x + y + z$$?

 

[Quinzio]

thought I understood equations of planes in R3 and their intersections, but apparently not. I'm very confused by what seems to be a basic problem:

find a vector equation for the line of intersection of x + y + z= 0 and x + z = 0.

Is x + z= 0 still a plane even though it doesn't have the form Ax + By + Cz = D?

I notice that if you set the two equations equal to each other you find that y = 1. Does this mean that the planes intersect on a line where y =1 and all x coordinates are equal to negative z coordinates? Thanks!

As a hint you have to use vectors and vector products.

 

[HallsofIvy]

No, you don't have to use vectors and vector products. (Except that the problem specifically asked for a vector as solution. If it had not you could write the line as parametric equations.) But I see no reason for "vector products".