Planes in

You are probably familiar with the expression “two points determine a line.” This means that given two distinct points, there is exactly one line that passes through both of them. There are other ways to describe a line. For example, as we saw in Parametric Equations of Lines, a single point together with a direction vector also determine a line.

You are probably also familiar with the fact that three non-collinear points determine a plane. (This is why photographers use tripods for stability, while four-legged chairs often wobble!) Is there another way to determine a plane?

The diagram below shows several planes perpendicular to vector . In fact, there are infinitely many such planes. Vector does not determine a plane, but if we know of a point contained in the plane, then together, and describe a unique plane.

Given a point and a nonzero vector , there is a unique plane through with normal .

This fact can be used to give a very simple description of a plane. Observe that a point lies on this plane if and only if the vector is orthogonal to (i.e. lies in the plane if and only if ).

Let . By “head-tail” formula (Formula form:headminustailr3), we have: . So, lies in the plane if and only if We summarize this result as a theorem.

As demonstrated in Example ex:planewithnormalvector, we can distribute coefficients , and of equation (eq:plane) as follows: Setting , shows that every plane with a normal vector has a linear equation of the form \begin{equation} \label{eq:eqofline} ax + by + cz = d \end{equation} for some constant . Conversely, the graph of this equation is a plane with as a normal vector (assuming that , , and are not all zero).

Linear Equations and their Graphs: From Lines to Hyperplanes

An equation of the form is a linear equation whose graph is a line in . If we solve for in terms of , we obtain a more familiar form of this equation, . The slope of the corresponding line is . Observe that a line perpendicular to this line has the slope . If we interpret as a horizontal “run” and as a vertical “rise”, we see that the vector is perpendicular to the line . You can use the following GeoGebra interactive to solidify your understanding of this.

The idea of coefficients in front of variables forming components of a normal vector should be very familiar to you. Recall that the graph of is a plane in with a normal vector .

Lines and planes may seem very different, but they are all graphs of linear equations, just in different dimensions. In general, an equation of the form is called a linear equation in variables. The graph of such an equation, for , is called a hyperplane. The vector in whose components are the coefficients is orthogonal to every vector in the hyperplane. Unfortunately, hyperplanes are impossible to see, but we can often use insights we gain from working with lines and planes and generalize them to the invisible world of higher dimensions.

Practice Problems

Problems prob:eqplane1-prob:eqplane3

Find an equation for each plane described below.

The plane has a normal vector and the plane passes through the point .

Answer:

The plane contains the point and is parallel to the plane described by .

Answer:

The plane contains the point and is parallel to the -plane.

Answer:

Problems prob:eqplane4-prob:eqplane6

How many planes satisfying each set of conditions are there?

Planes containing and with a normal vector .
0 1 Infinitely Many
Planes containing , , .
0 1 Infinitely Many
Planes containing , , .
0 1 Infinitely Many

Text Source

The text in this section is an adaptation of Section 4.2 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)

W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, p 233-234.

2024-09-11 17:57:37