# Finding the plane where the line lies on [closed]

Having a line and a normal orthogonal to it, how do I find the plane in which the normal will come out from and the line will lie on?

• $\mathbf{n}\cdot(\mathbf{x}-\mathbf{a})=0$ is normal plane equation where $\mathbf{n}$ is the normal, $\mathbf{x}=(x,y,z)^T$ and $\mathbf{a}$ is a point on the plane (you can take any point on the line for $\mathbf{a}$) Jul 2, 2020 at 12:40
• So,let $l$ be a line with a vector $u$ perpendicular to the line, you have to find a plane $\pi$ such that its orthogonal vector is $u$ and $l \subset \pi$, right? Jul 2, 2020 at 12:41
• @AlexeyBurdinwhat would T be? @psidaga exactly. Jul 2, 2020 at 12:43
• $T$ means transpose: $(x,y,z)^T=\pmatrix {x\\y\\z}$ Jul 2, 2020 at 12:44
• Ok, perfect. Are you in 3-dimensional space or in a general n-dimensional affine space? Jul 2, 2020 at 12:45

In the three-dimensional affine space $$\mathbb{A^3}_{\mathbb{R}}$$ (I'm assuming $$\mathbb{K}=\mathbb{R}$$) a line is defined, in its cartesian form, by the intersection of two planes.

Let $$l$$:$$\begin{cases} ax+by+cz+d=0 \\ a'x+b'y+c'z+d'=0 \end{cases}$$ and $$\textbf{u}$$ a vector orthogonal to the line.

We can consider the pencil of planes with support the line $$l$$ such that $$\Lambda_{\mu,\eta}: \mu (ax+by+cz+d) + \eta(a'x+b'y+c'z+d')=0$$
that can be written as $$x(\mu a + \eta a')+y(\mu b + \eta b')+z(\mu c + \eta c')+\mu d+\eta d'=0$$, where $$\textbf{v}=(\mu a + \eta a', \mu b + \eta b',\mu c + \eta c')$$ is the parametrization of its orthogonal vector.

So, in order to find the plane we only have to solve the system $$\textbf{v}= \rho \textbf{u}$$ (the two vectors have to be proportional).

• That solves my problem. But I was looking into the documentation of my programming language/library and found out that a plane can be generated by using 3 points that are on it as long as they are not forming a line. Probably very dummy question: is it possible to get multiple perpendiculars of the unit normal by just swapping x, y and z in different combinations? Jul 2, 2020 at 13:22
• Sorry, I forgot the terms d and d' (they were not really important for our problem). Now it's correct at all I think. Jul 2, 2020 at 13:24
• I'm not understanding if you are looking for the equation of the plane for those three points or for another thing... Jul 2, 2020 at 13:26
• Never mind. I got the problem solved with your main answer already. Thanks. Jul 2, 2020 at 13:37
• What is $\rho$ ? Jul 3, 2020 at 13:51