Vector Equations of a Plane

Introduction

The vector equation of a plane is a mathematical representation of a plane in 3D space. This equation is derived using vectors and can also be converted into its Cartesian form. Planes are fundamental in geometry and are widely used in physics, computer graphics, and engineering.

Vector Equation of a Plane

The vector equation of a plane is given by:

\[ \vec{r} = \vec{a} + \lambda \vec{b} + \mu \vec{c} \]

Here:

• \( \vec{r} \): The position vector of any point on the plane.
• \( \vec{a} \): The position vector of a known point on the plane.
• \( \vec{b} \) and \( \vec{c} \): Non-parallel vectors lying within the plane, also known as direction vectors.
• \( \lambda \) and \( \mu \): Scalars that allow the combination of \( \vec{b} \) and \( \vec{c} \) to span the plane.

Alternative Form: Normal Vector Equation

The equation of a plane can also be expressed in terms of a normal vector \( \vec{n} \), which is perpendicular to the plane:

\[ \vec{r} \cdot \vec{n} = \vec{a} \cdot \vec{n} \]

Here:

• \( \vec{r} \): The position vector of any point on the plane.
• \( \vec{a} \): The position vector of a known point on the plane.
• \( \vec{n} \): The normal vector to the plane.

This form is particularly useful for determining whether a point lies on the plane by substituting the point’s coordinates into the equation.

Cartesian Equation of a Plane

The vector equations can be converted into the Cartesian form of a plane:

\[ ax + by + cz = d \]

Here:

• \( a \), \( b \), \( c \): Components of the normal vector \( \vec{n} = [a, b, c] \).
• \( x, y, z \): Coordinates of a point on the plane.
• \( d \): A constant representing \( \vec{a} \cdot \vec{n} \), where \( \vec{a} \) is a known point on the plane.

Geometric Interpretation

• Normal Vector: The normal vector \( \vec{n} \) is perpendicular to the plane and uniquely defines its orientation.
• Direction Vectors: The vectors \( \vec{b} \) and \( \vec{c} \) span the plane and determine its extent.

Example Calculations

Example 1: Vector Equation of a Plane

Let \( \vec{a} = [1, 2, 3] \), \( \vec{b} = [2, 1, 0] \), and \( \vec{c} = [0, 1, -1] \). The vector equation of the plane is:

\[ \vec{r} = [1, 2, 3] + \lambda [2, 1, 0] + \mu [0, 1, -1] \]

Example 2: Normal Vector Equation

Given \( \vec{a} = [1, 2, 3] \) and \( \vec{n} = [2, -1, 1] \), the equation of the plane is:

\[ \vec{r} \cdot [2, -1, 1] = [1, 2, 3] \cdot [2, -1, 1] \]

\[ 2x – y + z = 2(1) – 1(2) + 1(3) = 3 \]

Thus, the Cartesian form is:

\[ 2x – y + z = 3 \]

Example 3: Cartesian Equation to Vector Form

Convert \( 3x + 2y – z = 6 \) into vector form:

• The normal vector is \( \vec{n} = [3, 2, -1] \).
• A point on the plane is \( [2, 0, 0] \), where \( x = 2 \), \( y = 0 \), and \( z = 0 \) satisfy the equation.

The vector equation is:

\[ \vec{r} = [2, 0, 0] + \lambda [2, -1, 0] + \mu [1, 1, 1] \]

Applications of Plane Equations

Equations of planes have wide-ranging applications, including:

• 3D Geometry: Analyzing shapes and intersections in space.
• Physics: Calculating forces, torques, and equilibrium conditions.
• Computer Graphics: Rendering and collision detection.