Planes

What is a Plane?

A plane is a flat, two-dimensional surface that extends infinitely in 3D space. Think of it like an endless sheet of paper floating in space, it has length and width, but no thickness.

How Do We Define a Plane?

A plane in 3D space can be defined in two main ways:

1. Using a Point and a Normal Vector (General)

The general equation of a plane is:

\[ \begin{gathered} ax+by+cz=d \\ \vec{n}\cdot(\vec{r}-\vec{r_0}) \end{gathered} \]

Here:

\(\vec{n}=⟨a,b,c⟩\) is the normal vector i.e. a vector that is perpendicular to the plane and a,b,c are the components of the vectors.
\(\vec{r} = ⟨x,y,z⟩\) is a variable point on the plane.
\(\vec{r_0} = ⟨x_0,y_0,z_0⟩\) is a known point on the plane.
d is a constant that shifts the plane away from the origin along the direction of the normal vector.

This form tells us: any point (x, y, z) that satisfies the equation lies on the plane.

Example:

\[ \begin{gathered} \vec{n}\cdot(\vec{r}-\vec{r_0}) = \langle 4, 5, 6 \rangle \cdot \langle x - 1, y - 2, z - 3 \rangle \\ 4(x - 1) + 5(y - 2) + 6(z - 3) = 0 \\ 4x + 5y + 6z = 32 \end{gathered} \]

This describes a plane in 3D space with normal vector (2,3,5).

2. Using Two Vectors (Parametric Form)

A plane can also be defined using two linearly independent vectors that lie in the plane. Generally used for visualizations, sampling etc. If \(v\) and \(w\) are such vectors, then all points on the plane can be described by:

To define a plane using vectors, you need:

A point \(\mathbf{p} = (x_0, y_0, z_0)\) that lies on the plane
Two linearly independent vectors \(v, w\) that lie in the plane

Then the parametric equation of the plane is:

\[ r(s,t)=p+sv+tw \]

Where:

\(s\) and \(t\) are real numbers (parameters)
\(r(s,t)\) gives all points on the plane

Example:

Let:

Point on the plane: \(p=(1,0,0)\)
Direction vectors: \(v=(2,2,0), w=(0,2,1)\)

Then the plane is:

\[ r(s,t)=(1,0,0)+s(2,2,0)+t(0,2,1) \]

Expanding:

\[ r(s,t)=(1+2s,2s+2t,t) \]

This gives all points on the plane by varying \(s\) and \(t\).

Why Are Planes Important?

Planes are fundamental in geometry, linear algebra, computer graphics, and physics. They help:

Visualize and solve systems of linear equations
Define surfaces, collisions, or boundaries
Represent flat objects like walls, screens, or layers in 3D models

References

https://www.youtube.com/watch?v=HjJ140TYbXQ