Equation of Plane

In three-dimensional space, a plane can be defined as a flat, two-dimensional surface that extends infinitely in all directions. The equation of a plane is a mathematical representation that describes all the points (x, y, z) that lie on the plane.

Vector Form of the Equation of a Plane

A plane can be defined using a point and a normal vector. The normal vector is perpendicular to the plane. If we have a point $P_0(x_0, y_0, z_0)$ and a normal vector $\vec{n} = (a, b, c)$, then the vector form of the equation of a plane is given by:

$$ \vec{n} \cdot (\vec{r} - \vec{r_0}) = 0 $$

where $\vec{r} = (x, y, z)$ is the position vector of any point on the plane, and $\vec{r_0} = (x_0, y_0, z_0)$ is the position vector of the given point $P_0$.

Expanding the dot product, we get:

$$ a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 $$

Scalar (Cartesian) Form of the Equation of a Plane

The scalar form of the equation of a plane is derived from the vector form and is written as:

$$ ax + by + cz + d = 0 $$

where $a$, $b$, and $c$ are the components of the normal vector $\vec{n}$, and $d$ is a scalar constant. The value of $d$ can be found by substituting the coordinates of the given point $P_0$ into the equation:

$$ d = - (ax_0 + by_0 + cz_0) $$

General Form of the Equation of a Plane

The general form of the equation of a plane is similar to the scalar form and is usually written as:

$$ Ax + By + Cz + D = 0 $$

where $A$, $B$, $C$ are the coefficients that correspond to the normal vector, and $D$ is the constant term.

Table of Differences and Important Points

Property	Vector Form	Scalar Form	General Form
Definition	Defined using a point and a normal vector	Derived from the vector form	Similar to scalar form, often used for simplicity
Equation	$\vec{n} \cdot (\vec{r} - \vec{r_0}) = 0$	$ax + by + cz + d = 0$	$Ax + By + Cz + D = 0$
Normal Vector	Explicitly used	Coefficients $a$, $b$, $c$ represent the normal vector	Coefficients $A$, $B$, $C$ represent the normal vector
Given Point	Required to find the equation	Used to find the constant $d$	Not explicitly used in the equation
Constant Term	Not present in vector form	$d = - (ax_0 + by_0 + cz_0)$	$D$ is the constant term

Examples

Example 1: Finding the Equation of a Plane

Given a point $P_0(1, -2, 3)$ and a normal vector $\vec{n} = (4, -5, 6)$, find the equation of the plane.

Solution:

Using the vector form, we have:

$$ 4(x - 1) - 5(y + 2) + 6(z - 3) = 0 $$

Expanding and simplifying, we get the scalar form:

$$ 4x - 5y + 6z - 23 = 0 $$

Example 2: Determining if a Point Lies on a Plane

Determine if the point $Q(2, -1, 4)$ lies on the plane $2x - 3y + z - 6 = 0$.

Solution:

Substitute the coordinates of point $Q$ into the equation of the plane:

$$ 2(2) - 3(-1) + 1(4) - 6 = 4 + 3 + 4 - 6 = 5 $$

Since the result is not zero, point $Q$ does not lie on the plane.

Understanding the equation of a plane is crucial for solving problems in three-dimensional geometry, such as finding the intersection of planes, distances from points to planes, and angles between planes.