]> 5.4 Representations of a Plane in 3 Dimensions

## 5.4 Representations of a Plane in 3 Dimensions

We now address the question: what is the relation between the different ways to describe a plane --- by points, one point and a vector, or by an equation?

Suppose the points $P 1 , P 2 , P 3$ lie in plane $Q$ and they are not all on a line.

Then the vectors $P 1 P 2 ⟶$ , and $P 1 P 3 ⟶$ have direction in $Q$ and an arbitrary point in $Q$ will have the coordinates of $O P 1 ⟶ + s * P 1 P 2 ⟶ + t * P 1 P 3 ⟶$ for some pair of values $( s , t )$ .

This is called a "parametric" representation of the plane with parameters $s$ and $t$ .

( $s$ and $t$ can be considered the components of the point in the plane in the basis given by $P 1 P 2 ⟶$ and $P 1 P 3 ⟶$ with origin $P 1$ .)

You can compute a normal to $Q$ by taking the cross product $P 1 P 2 ⟶ × P 1 P 3 ⟶$ .

We abbreviate by defining

$N ⟶ = P 1 P 2 ⟶ × P 1 P 3 ⟶ = O P 2 ⟶ × O P 3 ⟶ + O P 3 ⟶ × O P 1 ⟶ + O P 1 ⟶ × O P 2 ⟶$

and so the equation of the plane becomes

$N ⟶ · O P ⟶ = N ⟶ · O P 1 ⟶$

where $O P ⟶ = ( x , y , z )$ .

You can write this out explicitly as

$N x x + N y y + N z z = ( N , P 1 ) = ( N x P 1 x + N y P 1 y + N z P 1 z$

It is common, but not necessary to "normalize" $N ⟶$ , that is to replace it by $n ⟶$ with $n ⟶ = N ⟶ | N ⟶ |$ (remember $| N ⟶ | = ( N ⟶ · N ⟶ ) 1 / 2$ ) here.

In practice, planes are usually described by a normal vector, like $n ⟶$ here, and by a point in it.

We started with three points, and obtained a parametric representation of the plane from them. We then found an equation describing the plane from that representation.

If we can go from the description of $Q$ by this equation back to three points in $Q$ , we will be able to go all the way around the circle and find any representation of $Q$ from any other.

There are an infinite number of points in $Q$ and choosing three of them requires making arbitrary decisions to single out three of them.

If $N ⟶$ has all three of its components non-zero we can set each pair of variables to zero and solve for the third one. Then the three points will be

$( 0 , 0 , N ⟶ · O P 1 ⟶ N z ) , ( 0 , N ⟶ · O P 1 ⟶ N y , 0 )$

and

$( N ⟶ · O P 1 ⟶ N x , 0 , 0 )$

which are the points at which the plane meets the three axes.

From these points you can go around the circle again, and determine any representation of $Q$ .

In the applet here you can enter three arbitrary points, and it will find and picture the plane, show $N ⟶ , ( N ⟶ , O P 1 ⟶ )$ , and the parametric representation of points on it. You can do all of these things except making the picture, yourself.

Exercises:

5.2 Write down the equation for the default plane in this applet, and find the three points in that plane which have two 0 coordinates each.

5.3 Start with three random points and go through this procedure to find $N ⟶$ and the points where the plane meets the axes.

5.4 Set up a spreadsheet that does this whenever $N ⟶$ has all its components non-zero.

When $N z$ is not zero, we can solve the equation of the plane, $N ⟶ · O P ⟶ = N ⟶ · O P 1 ⟶$ for $z$ in terms of $x$ and $y$ , getting

$z = − N x N z x − N y N z y + N ⟶ · O P 1 ⟶ N z$

The coefficients of $x$ and $y$ here are particularly interesting to us. If you fix $y$ , then our three dimensional space becomes a plane. $− N x N z$ then represents the slope of the line that is the intersection of our plane and the plane described by the equation: $y = constant$ . The same statement holds after interchanging $x$ and $y$ in this one.

5.5 Find the two slopes (with $y$ fixed and $x$ fixed) h for the planes you describe in exercises 5.3 and 5.4.