graphing 3D question: z = x - y is a sloped plane. Then is 2 = x - y a line on that plane?

Vol

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Hi. I am studying the method of characteristics for PDEs and am stuck on the graphing. z = x - y is a sloped plane. Then is 2 = x - y a line on that plane? :(
 

HallsofIvy

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Hi. I am studying the method of characteristics for PDEs and am stuck on the graphing. z = x - y is a sloped plane. Then is 2 = x - y a line on that plane? :(
Assuming we are still working in 3 dimensions, yes 2= x- y is a line on that plane. It is the line where the plane z= 2 intersects the plane z= x- y.

If z had never been mentioned, we would assume it was a line in the two dimensional xy- plane. That would be the same as projecting the three dimensional line onto the xy-plane.
 

Vol

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Assuming we are still working in 3 dimensions, yes 2= x- y is a line on that plane. It is the line where the plane z= 2 intersects the plane z= x- y.

If z had never been mentioned, we would assume it was a line in the two dimensional xy- plane. That would be the same as projecting the three dimensional line onto the xy-plane.
Oh, OK. That is what I thought. But I could not find an explanation. The 3D graphers online showed weird things. Like a vertical plane. The another showed a plane at z = 2. None of them showed a line.
 

Dr.Peterson

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Oh, OK. That is what I thought. But I could not find an explanation. The 3D graphers online showed weird things. Like a vertical plane. The another showed a plane at z = 2. None of them showed a line.
The equation 2 = x - y represents a plane; its intersection with the plane z = x - y is a line in the plane z = 2.
 
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