» Required Reading » Table of Contents » Chapter 5 5.2 Representations of a Plane
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A single linear equation can be used to solve for one variable in terms of the others. It reduces the dimension of the set of its solutions by 1.

Thus the points in a plane in 3 dimensions will be the solutions of one  linear equation. In two dimensions one linear equation determines a line, while in three dimensions two equations are needed to determine a line.

A linear equation of the form ax + by + cz = d can be written as the dot product (vr) = d where v is the vector (a, b, c) and r is (x, y, z).

Thus solutions to it all have the same value of their component in the direction of v, and are not determined in directions perpendicular to, normal to, or orthogonal to v. (all these words mean the same thing). Thus v is normal to a vector pointing from one solution to another.

A line can be characterized by giving the coordinates of two points on it (in any space) and a plane by giving the coordinates of three points that don’t all lie on a single line.

Further we can describe a line by giving any point on it, and a vector that points in its direction.

Likewise we can describe the points on a plane by giving a point and two linearly independent vectors starting from that point that lead to other points in the plane.

We first 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 P₁, P_2, P₃ lie in plane Q and they are not all on a line.

Then the vectors P₂ - P₁, and P₃ - P₁ have direction in Q and an arbitrary point in Q will have the coordinates of

P₁ + s (P₂ - P₁) + t (P₃ - P₁) 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 (P₂ - P₁) and (P₃ - P₁) with origin P₁.)

You can compute a normal to Q by taking the cross product (P₂ - P₁)(P₃ - P₁).

We abbreviate by defining

N = (P₂ - P₁)(P₃ - P₁)

and so the equation of the plane becomes

Nr = NP_1.

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

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 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

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

Exercises:

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

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

When N_z jis not zero, we can solve the equation of the plane, Nr = NP₁ for z in terms of x and y, getting

The coefficients of x and y here are particularly interesting to us. They represent respectively, the slope of z in the x direction (keeping y fixed) and in the y direction (keeping x fixed.)
We now ask: how do we describe a line L?