Simple question about parametric equations of a plane in 3D

In summary, a plane in 3D with the equation ##z=2## has two parameters, ##x## and ##y##, and can be graphed as a horizontal plane in 3D. However, when graphed in Wolfram Alpha, it appears as a line because the software does not have an easy way to show a plane in R^3. Introducing an additional variable, such as ##t##, is not necessary.
  • #1
ainster31
158
1
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
 
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  • #2
hi ainster31! :smile:
ainster31 said:
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 :wink:

your plane is z = 2, x = t, y = u​
 
  • #3
tiny-tim said:
hi ainster31! :smile:line: one parameter

plane: two parameters :wink:

your plane is z = 2, x = t, y = u​

Hmm... what about 3 parameters? What would that result in? A filled 3D cube, right?
 
  • #4
If x, y, and z are arbitrary, you get the entire space (all of R3).
 
  • #5
ainster31 said:
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! :smile:​
 
  • #6
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.
 

Related to Simple question about parametric equations of a plane in 3D

1. What are parametric equations of a plane in 3D?

Parametric equations of a plane in 3D are mathematical equations that represent a plane in three-dimensional space. They are expressed in terms of two independent variables, usually denoted by u and v, which correspond to the horizontal and vertical directions on the plane. These equations help describe the position and orientation of the plane in space.

2. How are parametric equations of a plane different from standard equations of a plane?

Parametric equations of a plane allow for a more versatile and flexible representation of a plane in 3D space. Unlike standard equations of a plane, which are typically written in the form Ax + By + Cz + D = 0, parametric equations use two independent variables to describe the position and orientation of the plane. This allows for more complex shapes and surfaces to be represented.

3. How do you graph parametric equations of a plane in 3D?

To graph parametric equations of a plane in 3D, you can use a graphing calculator or a computer program that supports 3D graphing. First, plot the points generated by the parametric equations and connect them to form a surface. You can also use the parametric equations to create a parametric curve on the surface of the plane.

4. What is the purpose of using parametric equations of a plane in 3D?

The purpose of using parametric equations of a plane in 3D is to provide a more versatile and powerful representation of a plane in three-dimensional space. These equations allow for more complex and dynamic shapes to be represented, making them useful in fields such as mathematics, engineering, and computer graphics.

5. How do parametric equations of a plane relate to vectors?

Parametric equations of a plane in 3D are closely related to vectors. The independent variables u and v correspond to the components of a vector that lies on the plane. The direction of this vector, together with the position of the plane, can be determined from the parametric equations. Additionally, the cross product of two vectors on the plane can be used to find the normal vector, which can then be used to write the standard equation of the plane.

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