Plane in Normal Form (Vector Form)

In 3D geometry, a plane can be represented in various forms, one of which is the normal form (also known as the vector form). This form is particularly useful because it uses a normal vector to define the orientation of the plane.

Understanding the Normal Form

The normal form of a plane is based on the concept that any plane can be defined by a point through which it passes and a vector that is perpendicular to the plane, known as the normal vector.

The equation of a plane in normal form is given by:

$$ \vec{r} \cdot \vec{n} = d $$

where:

• $\vec{r}$ is the position vector of any point on the plane,
• $\vec{n}$ is the normal vector to the plane,
• $d$ is the perpendicular distance from the origin to the plane.

Breaking Down the Equation

• Position Vector $\vec{r}$: This is a vector that represents the position of a point in space relative to the origin. In the context of the plane equation, it represents the position of any point that lies on the plane.
• Normal Vector $\vec{n}$: This is a vector that is perpendicular to the plane. It defines the orientation of the plane in space. The normal vector is often normalized (i.e., made to have a unit length) for simplicity.
• Dot Product: The dot product (also known as the scalar product) is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In geometric terms, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.
• Perpendicular Distance $d$: This is the shortest distance from the origin to the plane. It is a scalar quantity and can be positive or negative, depending on which side of the origin the plane is located.

Table of Important Points

Point	Description
Normal Vector	A vector perpendicular to the plane, denoted as $\vec{n}$.
Position Vector	A vector representing the position of a point on the plane, denoted as $\vec{r}$.
Dot Product	An operation that multiplies two vectors, yielding a scalar, denoted as $\vec{r} \cdot \vec{n}$.
Perpendicular Distance	The shortest distance from the origin to the plane, denoted as $d$.

Formulas

• Dot Product: $\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z$
• Equation of Plane in Normal Form: $\vec{r} \cdot \vec{n} = d$

Examples

Example 1: Finding the Equation of a Plane

Given a normal vector $\vec{n} = \begin{bmatrix} 2 \ 3 \ -1 \end{bmatrix}$ and a point $P(1, 2, 3)$ on the plane, find the equation of the plane in normal form.

Solution:

First, we find the position vector of point $P$, which is $\vec{r}_P = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}$.

The perpendicular distance $d$ from the origin to the plane can be found by taking the dot product of $\vec{n}$ and $\vec{r}_P$:

$$ d = \vec{n} \cdot \vec{r}_P = \begin{bmatrix} 2 \ 3 \ -1 \end{bmatrix} \cdot \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix} = (2)(1) + (3)(2) + (-1)(3) = 2 + 6 - 3 = 5 $$

Therefore, the equation of the plane in normal form is:

$$ \vec{r} \cdot \begin{bmatrix} 2 \ 3 \ -1 \end{bmatrix} = 5 $$

Example 2: Determining if a Point Lies on a Plane

Given the plane equation $\vec{r} \cdot \begin{bmatrix} 1 \ -2 \ 2 \end{bmatrix} = 4$ and a point $Q(3, -1, 2)$, determine if $Q$ lies on the plane.

Solution:

Find the position vector of point $Q$, which is $\vec{r}_Q = \begin{bmatrix} 3 \ -1 \ 2 \end{bmatrix}$.

Substitute $\vec{r}_Q$ into the plane equation:

$$ \vec{r}_Q \cdot \begin{bmatrix} 1 \ -2 \ 2 \end{bmatrix} = \begin{bmatrix} 3 \ -1 \ 2 \end{bmatrix} \cdot \begin{bmatrix} 1 \ -2 \ 2 \end{bmatrix} = (3)(1) + (-1)(-2) + (2)(2) = 3 + 2 + 4 = 9 $$

Since $9 \neq 4$, point $Q$ does not lie on the plane.

By understanding the normal form of a plane, we can easily describe its orientation in space, find its equation, and determine the relationship between the plane and other geometric entities, such as points and lines. This form is particularly useful in vector calculus, physics, and engineering applications.