What is the Intersection of y=3 and z=5 in R^3?

  • Thread starter Saladsamurai
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In summary, the pair of equations y=3 and z=5 in R^3 represents the set of all points that satisfy both constraints, which is given by the intersection of two planes. These planes are parallel to the xz-plane and xy-plane, and the intersection results in a line unless the two equations represent the same plane. Noting the parallel orientation to coordinate planes may aid in visualizing the solution.
  • #1
Saladsamurai
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Homework Statement



In R^3, what does the pair of equations y=3 and z=5 represent? In other words, describe the set of points (x,y,z) such that y=3 and z=5.

The Attempt at a Solution



I drew out the individual planes y=3 and z=5. i am not sure how to describe the set of all points that satisfy both constraints?

I have to have the set of all points such that for ALL x-->y=3 and z=5.

Is that a surface? Or a line? I am thinking it has to be a line?

thoughts?
 
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  • #2
It might help to be a little more specific about the planes..

y = 3 is a plane, and it is parallel to the xz-plane.
z = 5 is a plane, and it is parallel to the xy-plane.

So the set of all (x,y,z) such that y=3 and z=5 is given by the intersection of these two planes, and since one is parallel to the xz-plane and the other is parallel to the xy-plane, then the intersection of y=3 and z=5 is a ______
 
  • #3
line!

Though, I am not sure why it is so important to note that each line is parallel to a coordinate plane? The intersection of 'any' 2 planes is a line, correct? No matter what their orientation is?

Thanks!
 
  • #4
well.. it's been a while since I've thought about things like that, but yeah, you're right. unless it's a trivial case where the two equations represent the same plane. either way, noting they're parallel to coordinate planes might help you visualize it better, but if you drew a picture anyway, it sounds like you're all set.
 
  • #5
I see. It has been a while for me too, so I am right there with you :smile: Thanks for chiming in xeno_gear!
 

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