Planes along with points and lines form the building blocks of Geometry. These are named as undefined terms as they are explained only using examples and descriptions and not with definitions. In algebra you are familiar with the coordinate plane on which points are plotted using ordered pairs and lines are represented as equations. Thus, a plane can be described as a flat surface made up of infinitely many points extending in all directions without thickness.Planes.


A plane can be represented by a capital letter like Plane X. It can also be represented using at least three non collinear points that lie on the plane, like plane ABC.

Planes are represented by algebraic equations in 3D-Geometry. The 3D-space is illustrated using 3 coordinate axes, x, y and z. The coordinate planes are names as xy, yz and xz planes named after the axes contained by the planes as shown in the diagram.

3D Plane Axes


The 3-D equation to XY - plane is z = 0.
Similarly the respective equations to YZ and XZ planes are x = 0 and y = 0 .
The equations of planes parallel to coordinate planes are as follows:
z = k is the equation of a plane parallel to xy - plane at a signed distance k from it.
x = k is the equation of a plane parallel to yz - plane at a signed distance k from it.
y = k is the equation of a plane parallel to xz - plane at a signed distance k from it.
A first degree equation in variables x, y and z represent a general plane in 3-D space.
ax + by + cz + d = 0.

Different formulas both vector and algebraic are available to write the equation of a plane based on given conditions.

1. Equation to a plane containing the point P (x0, y0, z0) and perpendicular to the direction $\overrightarrow{n}$ =
The vector form of the equation is n . r = n . r0. where r0 is the position vector 0, y0, z0>.
The 3-D form of the equation is a(x - x0) + b(y - y0) + c(z - z0) = 0.
Combining the constant terms, the equation to the plane can be written as ax + by + cz + d = 0.
2. Equation to a plane passing through a given point P(x0, y0, z0) and parallel to two vectors
$\overrightarrow{u}$ = 1, m1, n1> and $\overrightarrow{v}$ = < l2, m2, n2>
The vector form of the equation is
$\overrightarrow{r}$ = $\overrightarrow{a}$ + s $\overrightarrow{u}$ + t $\overrightarrow{v}$
where $\overrightarrow{a}$ is the position vector of P.
The algebraic form of the equation is $\begin{vmatrix}
x-x_{0}&y-y_{0} & z-z_{0}\\
l_{1}& m_{1} & n_{1}\\
l_{2}& m_{2} & n_{2}
\end{vmatrix}$ = 0
3. Equation of a plane containing two given points P(x1, y1, z1) and Q(x2, y2, z2) and parallel to a given vector $\overrightarrow{v}$ = .
The vector form of the equation is $\overrightarrow{r}=(1-s)\overrightarrow{a}+s\overrightarrow{b}+t\overrightarrow{v}$
where $\overrightarrow{a}$ and $\overrightarrow{b}$ are position vectors of the points P and Q and t and s are
parameters.
The algebraic form of the equation is $\begin{vmatrix}
x-x_{1} & y-y_{1} &z-z_{1} \\
x-x_{2} &y-y_{2} &z-z_{2} \\
l & m & n
\end{vmatrix}$ = 0
4. The equation of the plane passing through three given points P(x1, y1, z1), Q(x2, y2, z2) and R(x3, y3, z3).
The vector form of the equation is
$\overrightarrow{r}$ = $\overrightarrow{a}$ + s($\overrightarrow{b}$ - $\overrightarrow{a}$) + t($\overrightarrow{c}$ - $\overrightarrow{a}$) where $\overrightarrow{a}$, $\overrightarrow{b}$ and $\overrightarrow{c}$ are the position vectors
of the points P, Q and R.
The algebraic form of the equation is $\begin{vmatrix}
x-x_{1} & y-y_{1} &z-z_{1} \\
x-x_{2}& y-y_{2} &z-z_{2} \\
x-x_{3}&y-y_{3} & z-z_{3}
\end{vmatrix}$ = 0