Simple question about parametric equations of a plane in 3D

[ainster31]

I'm quite rusty in Linear Algebra.

If you have a plane in 3D with the equation $z=2$, what does $x$ and $y$ equal? Does $x=t$ and $y=t$?

Because if I graph that in Wolfram Alpha, I don't get a horizontal plane in 3D at $z=2$: http://www.wolframalpha.com/input/?i=graph+z=2,x=t,y=t

 

[tiny-tim]

hi ainster31!

If you have a plane in 3D with the equation $z=2$, what does $x$ and $y$ equal? Does $x=t$ and $y=t$?

line: one parameter

plane: two parameters

your plane is z = 2, x = t, y = u​

 

[ainster31]

hi ainster31! line: one parameter

plane: two parameters

your plane is z = 2, x = t, y = u​

Hmm... what about 3 parameters? What would that result in? A filled 3D cube, right?

 

[Mark44]

If x, y, and z are arbitrary, you get the entire space (all of R³).

 

[tiny-tim]

Hmm... what about 3 parameters? What would that result in? A filled 3D cube, right?

3 dimensions: 3 parameters …

n dimensions: n parameters …

that's very nearly a definition of dimensions! ​

 

[TheOldHag]

z=2 is the equation of a plane in R^3. x and y range over R since they are not specified. So you essentially end up with an x,y plane. In wolfram alpha they just show a line since there doesn't appear to be an easy way to tell it you want R^3. Your initial assumption was correct and variable t shouldn't be introduced.