Describe a plane

Picture the coordinate planes

Choose the plane description from the information given: a point and two independent directions, or a point and a nonzero normal. Keep that distinction when converting forms. A normal is perpendicular to the plane; it is not a direction along it.

A plane is a flat, two-dimensional surface extending indefinitely.

In three-dimensional coordinates, each pair of axes determines a coordinate plane: the $xy$-plane ($z=0$), $xz$-plane ($y=0$), and $yz$-plane ($x=0$).

The source room analogy places the floor in the $xy$-plane, the left wall in the $xz$-plane and the right wall in the $yz$-plane, using its chosen axes.

The walls and floor are finite pieces of these planes; there are also planes in other orientations.

Move within a plane

Choose a point A with position vector $\mathbf a$ and two nonzero, nonparallel directions $\mathbf b,\mathbf c$ parallel to the plane. For any point R in the plane,

$$\begin{aligned}\overrightarrow{AR}&=\lambda\mathbf b+\mu\mathbf c,\\\overrightarrow{OR}-\overrightarrow{OA}&=\lambda\mathbf b+\mu\mathbf c,\\\mathbf r&=\mathbf a+\lambda\mathbf b+\mu\mathbf c,\quad\lambda,\mu\in\mathbb R.\end{aligned}$$

The drawn parallelogram is a finite representation of an infinite plane; any patch shape can be used. Real parameters are not restricted to its edges. Changing the base point or independent directions gives another equation of the same plane. Parallel directions span only a line. The source QR code links to a parametric-plane activity; this source pointer is retained, without claiming a review of the external activity.

Use a normal instead of two directions

For A and B in a plane with nonzero normal $\mathbf n$, the in-plane displacement is perpendicular to $\mathbf n$:

$$\begin{aligned}\overrightarrow{AB}\cdot\mathbf n&=0,\\(\mathbf b-\mathbf a)\cdot\mathbf n&=0,\\\mathbf b\cdot\mathbf n&=\mathbf a\cdot\mathbf n=d.\end{aligned}$$

The same value is obtained at any point R, giving the scalar-product form

$$\mathbf r\cdot\mathbf n=\mathbf a\cdot\mathbf n.$$

For two independent in-plane directions $\mathbf p,\mathbf q$, a normal is $\mathbf p\times\mathbf q$. If the plane passes through the origin, $\mathbf a\cdot\mathbf n=0$. The converse also holds when $\mathbf n\ne\mathbf0$.

The source lists four ways to determine a unique plane: a nonzero normal and a point; two nonparallel directions and a point; two intersecting, nonparallel lines; or three non-collinear points. A zero normal gives no plane constraint, and three collinear points do not determine a unique plane.

For a line-plane angle, first compare the line direction with the plane normal, then take the complementary acute angle. For two planes, compare their normals.

Convert between plane equations

Write $\mathbf r=(x,y,z)$ and the nonzero normal $\mathbf n=(n_1,n_2,n_3)$. Expanding $\mathbf r\cdot\mathbf n=d$ gives

$$n_1x+n_2y+n_3z=d.$$

These are equivalent forms of the same plane; choose the one convenient for the task. The source summary gives the conversion routes:

• Parametric $\mathbf r=\mathbf a+\lambda\mathbf m_1+\mu\mathbf m_2$: find $\mathbf n=\mathbf m_1\times\mathbf m_2$, then $d=\mathbf a\cdot\mathbf n$.
• Scalar product $\mathbf r\cdot\mathbf n=d$: expand the dot product to obtain the Cartesian equation.
• Cartesian $n_1x+n_2y+n_3z=d$: read off $\mathbf n=(n_1,n_2,n_3)$ to recover the scalar-product form.

The directions must be independent and the normal nonzero. Multiplying both the normal and $d$ by the same nonzero scalar preserves the plane. Scaling only one side generally changes it.

Three forms of each coordinate plane

All parameters below are real.

$xy$

Parametric vector form:

$$\mathbf r=\lambda\mathbf i+\mu\mathbf j$$

Scalar-product form:

$$\mathbf r\cdot\mathbf k=0$$

Cartesian form:

$$z=0$$

$yz$

Parametric vector form:

$$\mathbf r=\lambda\mathbf j+\mu\mathbf k$$

Scalar-product form:

$$\mathbf r\cdot\mathbf i=0$$

Cartesian form:

$$x=0$$

$xz$

Parametric vector form:

$$\mathbf r=\lambda\mathbf i+\mu\mathbf k$$

Scalar-product form:

$$\mathbf r\cdot\mathbf j=0$$

Cartesian form:

$$y=0$$