Planes and Points

Topics: Geometry - Plane - Point - 3D Space

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Planes and points can be related in several ways.

Plane that Passes Through the Origin §

When $P=(0,0,0)$ (the origin), then the plane takes the vectorial representation:

$$\Pi :=\{tu+sv:t,s\in \mathbb{R}\}$$

We call this the plane that passes through the origin and is generated by the vectors $u$ and $v$.

Does a Point Exist in a Plane? §

To see if $q=(q_1,q_2,q_3)$ is in a given plane $\Pi =\{P+tu+sv:t,s\in \mathbb{R}\}$, then we can try to solve the equation system:

$$\{\begin{array}{l}{q}_1=P_1+s{u}_1+t{v}_1\\ q_2=P_2+s{u}_2+t{v}_2\\ q_3=P_3+s{u}_3+t{v}_3\end{array}$$

(remember we can write $\Pi =\{P_1+s{u}_1+t{v}_1,P_2+s{u}_2+t{v}_2,P_3+s{u}_3+t{v}_3):t,s\in \mathbb{R})$)

If the system can be solved, then the point exists in the plane. If it can’t be solved, then it doesn’t exist in the plane.

Plane that Passes Through 3 Points §

Given 3 points $p=(x_1,y_1,z_1)$, $q=(x_2,y_2,z_2)$, $r=(x_3,y_3,z_3)$, we can find the plane that passes through these three points by doing the following:

1. Choosing one point as the base point. Let’s say we choose $p$
2. Calculating the direction vectors by subtracting the base point from the other points:
   • $u=r-p=p-r$
   • $v=q-p=p-q$
3. We can use the base point and these two vectors to represent the plane that passes through these three points