Equations of Planes

Section 3.3 Equations of Planes

Subsection 3.3.1 Dot Products and Loci

Loci are the solutions to one of more linear equations. These linear equation can be re-interpreted and expressed via via dot products. Consider, again, the general linear equation in \(\RR^n\text{.}\)

\begin{equation*} a_1 u_1 + a_2 u_2 + \ldots + a_n u_n = c \end{equation*}

Think of the variables \(u_i\) as the components of a vector \(u \in \RR^n\text{.}\) There also are \(n\) scalars \(a_i\text{,}\) which I can likewise treat as components of the vector \(a \in \RR^n\text{.}\) The vector \(u\) is variable; the vector \(a\) is constant. Then I can re-write the linear equation using these two vectors.

\begin{equation*} a_1 u_1 + a_2 u_2 + \ldots + a_n u_n = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} \cdot \begin{pmatrix} u_1 \\ u_2 \\ \vdots \\ u_n \end{pmatrix} = a \cdot u = c \end{equation*}

In this way, a linear equation specifies that the dot product result of a variable vector \(u\) with a fixed vector \(a\) must have the result \(c\text{.}\) In this light, an affine plane in \(\RR^3\) is given by the equation

\begin{equation*} \begin{pmatrix} x \\ y \\ z \end{pmatrix} \cdot \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} = c\text{.} \end{equation*}

This plane is precisely all vectors whose dot product with the vector \(\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\) is the fixed number \(c\text{.}\) If \(c=0\text{,}\) the equation become

\begin{equation*} \begin{pmatrix} x \\ y \\ z \end{pmatrix} \cdot \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} = 0\text{.} \end{equation*}

A dot product of zero indicates that the two inputs to the dot product are perpendicular. Therefore, a plane through the origin is the set of all vectors which are perpendicular to a fixed vector \(\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\text{.}\)

Definition 3.3.1.

Let \(P\) be a plane in \(\RR^3\) determined by the equation

\begin{equation*} \begin{pmatrix} x \\ y \\ z \end{pmatrix} \cdot \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} = c\text{.} \end{equation*}

The vector \(\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\) is called the normal to the plane. Let \(H\) be a hyperplane in \(\RR^n\) determined by the equivalent equation in \(\RR^n\text{.}\)

\begin{equation*} u \cdot a = \begin{pmatrix} u_1 \\ u_2 \\ \vdots \\ u_n \end{pmatrix} \cdot \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} = c \end{equation*}

The vector \(a\) is called the normal to the hyperplane.

If \(c=0\text{,}\) the plane or hyperplane is perpendicular to its normal. This notion of orthogonality still works when \(c \neq 0\text{.}\) In this case, the normal is a local perpendicular direction from any point on the affine plane. Treating any such point as a local origin, the normal points in a direction perpendicular to all the local direction vectors which lie on the plane.

Subsection 3.3.2 An Algorithm for Equations of Planes

Now I can build a general process for finding the equation of a plane in \(\RR^3\text{.}\) Any time I have a point \(p\) on the plane and two local direction vectors \(u\) and \(v\) which remain on the plane, I can find a normal to the plane by calculating \(u \times v\text{.}\) Then I can find the equation of the plane by calculating the dot product \(p \cdot (u \times v)\) to find the constant \(c\text{.}\)

\begin{equation*} \begin{pmatrix} x \\ y \\ z \end{pmatrix} \cdot (u \times v) = c \end{equation*}

If I am given three points on a plane (\(p\text{,}\) \(q\) and \(r\)), then I can use \(p\) as the local origin and construct the local direction vectors as \(q-p\) and \(r-p\text{.}\) The normal is \((q-p) \times (r-p)\text{.}\) In this way, I can construct the equation of a plane given three points or a single point and two local directions.

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