General Equation of a Plane

In three-dimensional geometry, a plane can be defined as a flat, two-dimensional surface that extends infinitely in all directions. The general equation of a plane is a linear equation that represents all the points lying on the plane. This equation can be derived in several ways, depending on the information available.

Vector Form

If we know a point ( \mathbf{A}(x_1, y_1, z_1) ) on the plane and a normal vector ( \mathbf{n} = (a, b, c) ) perpendicular to the plane, the vector equation of the plane is given by:

[ (\mathbf{r} - \mathbf{A}) \cdot \mathbf{n} = 0 ]

where ( \mathbf{r} = (x, y, z) ) is the position vector of any point on the plane.

Expanding the dot product, we get:

[ a(x - x_1) + b(y - y_1) + c(z - z_1) = 0 ]

Scalar (Cartesian) Form

The scalar form of the equation of a plane is derived from the vector form and is given by:

[ ax + by + cz + d = 0 ]

where ( a, b, ) and ( c ) are the components of the normal vector to the plane, and ( d ) is the scalar constant.

Point-Normal Form

If we know a point ( \mathbf{A}(x_1, y_1, z_1) ) on the plane and a normal vector ( \mathbf{n} = (a, b, c) ), the point-normal form of the equation of the plane is:

[ a(x - x_1) + b(y - y_1) + c(z - z_1) = 0 ]

Intercept Form

If a plane cuts the x, y, and z axes at points ( A(a, 0, 0) ), ( B(0, b, 0) ), and ( C(0, 0, c) ) respectively, the intercept form of the equation of the plane is:

[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 ]

Differences and Important Points

Here is a table summarizing the different forms of the equation of a plane and their important points:

Form	Equation	Important Points
Vector Form	( (\mathbf{r} - \mathbf{A}) \cdot \mathbf{n} = 0 )	Requires a known point and a normal vector.
Scalar Form	( ax + by + cz + d = 0 )	General form, easy to use for calculations.
Point-Normal Form	( a(x - x_1) + b(y - y_1) + c(z - z_1) = 0 )	Directly uses a point and the normal vector.
Intercept Form	( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 )	Useful when the plane's intercepts with the axes are known.

Examples

Example 1: Finding the Equation of a Plane

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

Solution:

Using the point-normal form:

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

Expanding and simplifying:

[ 4x - 4 - 5y + 10 + 6z - 18 = 0 ]

[ 4x - 5y + 6z - 12 = 0 ]

This is the scalar form of the equation of the plane.

Example 2: Finding the Equation of a Plane in Intercept Form

A plane intercepts the x-axis at 4, the y-axis at -3, and the z-axis at 2. Find the equation of the plane.

Solution:

Using the intercept form:

[ \frac{x}{4} + \frac{y}{-3} + \frac{z}{2} = 1 ]

Multiplying through by the common denominator, 12:

[ 3x - 4y + 6z = 12 ]

This is the scalar form of the equation of the plane.

Understanding the general equation of a plane is crucial for solving problems in 3D geometry, especially in fields such as physics, engineering, and computer graphics. Each form of the equation has its own applications and is used based on the given information and the context of the problem.