Sketching a Plane in Three Dimensional Space

Date: 11/15/2005 at 03:26:29
From: Peter
Subject: Three simultaneous equations with three unknowns

I know that an equation like ax + by + cz = p represents a plane in
three dimensions.  But I can't find any explanation on how to sketch
such an equation, like 2x + y + z = 3.  I'm particulary interested in
what part the number 3 plays in the positioning of the plane on the
xyz axes. 

Beyond sketching any one such equation, I'm also trying to sketch a
system of simultaneous equations to interpret them geometrically.  I
would like to see clearly how three simultaneous equations interact
with one another in the xyz axes.

I am aware that three simultaneous equations may have a unique 
solution, be inconsistent (having no solution), or be dependent 
(having infinite solutions).  But how do I sketch a system to show it
clearly on the xyz axes?

Date: 11/15/2005 at 08:16:04
From: Doctor Rick
Subject: Re: Three simultaneous equations with three unknowns

Hi, Peter.

It's really hard to sketch in three dimensions, since we have to do so 
on paper, making a 2-dimensional oblique view of the axes, and the 
third dimension can't be clearly distinguished.

Here is one thing that may help you visualize the planes.  Try 
plotting the points where a plane is intersected by each coordinate 
axis.  For example, the plane defined by 

  2x + y + z = 3

crosses the x axis at (3/2,0,0) - I just set y = z = 0 and solved the 
resulting equation 2x = 3.  Likewise it crosses the y axis at (0,3,0) 
and the z axis at (0,0,3).  Draw the three axes, mark the three points 
(A, B, and C below), and connect them to form a triangle.

            z
            |
           C*
            |
            |
            |
            |
            |
            |
            |
            |
            *
           /     \
        A /           \     B
         *                 *
        /                       \
       /                             y
      /
     /
    /
   x

This triangle lies in the plane; if you do the same for each plane, 
you'll get some idea of their relationships.  For instance, you'll see 
if they are parallel.  You may or may not see the line of intersection 
of two planes immediately, but if you extend the sides of the 
triangles until they intersect, you will.  For instance, consider the 
lines of intersection of two planes with the x-y plane: these will be 
the lines joining the x-intercept and the y-intercept of each plane.  
Where they intersect is one point on the intersection of the planes.  
Do the same with the y-z plane (or the x-z plane), and connect the two 
points you have found to make the line of intersection.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 11/16/2005 at 16:02:45
From: Peter
Subject: Thank you (Three simultaneous equations with three unknowns)

Dear Doctor Rick -

Thank you very much for your clear explanation and for your prompt
reply.  I appreciated it very much.