Line and plane (en)

28-06-2025

The Linear Algebra starts from basic geometry intuition, which continues to multivariate spaces. Now we will talk about line, plane and space.

# The line in plane and space

How to define a line in plane? Of course, through two points. But it’s not all. Exists are three more methods:

1. Parametric equation Let line $l$ set by two points $r_0 = (x_0; y_0)$ and $r_1 = (x_1;y_1)$. Observe that vector $a = r_0 - r_1$ in same direction with $l$. Therefore, any point on $l$ may define as $r = r_0 + ta$, where $t \in \mathbb{R}$ and full line $l$ way be with $-\infty < t < \infty$. This expression we call parametric equation:

   $$ l = \{r: r = r_o + at, \ t \in \mathbb{R} \} $$

2. Linear equation Let’s write the parametric equation using coordinates:

   $$ \begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}x_0\\y_0\end{bmatrix} + \begin{bmatrix}t \cdot a_1 \\ t \cdot a_2\end{bmatrix} \Rightarrow \begin{cases}x = x_0 + ta_1 \\ y = y_0 + ta_2\end{cases} $$

   Suppose that, $a_1 \neq 0$ and $a_2 \neq 0$ and solve $t$ from each equation: $\frac{x - x_0}{a_1} = \frac{y - y_0}{a_2} \Rightarrow a_2x - a_1y + (a_1y_0 - a_2x_0) = 0$. Since $a_1, a_2, x_0, x_1$ it’s a numbers, that mean we can define it as linear equation:

   $$ Ax + By + C = 0 $$

   where $A = a_2$, $B = -a_1$, $C = (a_1y_0 - a_2x_0)$

3. Normal and translation Last method. Let’s select random point on line $l$ with radius-vector $r_0$. Let’s draw a vector $n$ from point $r_0$, that is perpendicular line $l$. Then for any point $r$ on line $l$ vectors $r - r_0$ and $n$ will be perpendicular. Rewrite it with scalar product:

   $$ l = \{ r: \langle r - r_0;n\rangle = 0\} $$

   Of course, $r_0$ it can by any point on line $l$ and normal vector $n$ any vector perpendicular to line $l$. To find normal to line, we need direction vector $a = \begin{pmatrix}a_1 \\ a_2\end{pmatrix}$ from parametric equation and then normal vector will be: $n = \begin{bmatrix}-a_2\\ a_1\end{bmatrix}$ or $n’ = -n = \begin{bmatrix} a_2 \\ -a_1\end{bmatrix}$

Let’s resume and see the relationship between this methods. If line passes through two points $r_0 = (x_0; y_0)$ and $r_1 = (x_1; y_1)$, then his can be define of of the three methods:

• parametric equation : $$ r = r_0 + ta = \begin{bmatrix}x_0\\y_0\end{bmatrix} + t \cdot \begin{bmatrix}x_1 - x_0 \\ y_1 - y_0\end{bmatrix} = \begin{bmatrix}x_0\\y_0\end{bmatrix} + \begin{bmatrix}t \cdot a_1 \\ t \cdot a_2\end{bmatrix} $$
• linear equation: $$ Ax + By + C = \underbrace{a_2}_{A} x + \underbrace{(-a_1)}_{B} y + \underbrace{(a_1 y_0 - a_2 x_0)}_{C} = 0; $$
• normal and translation: $$ \langle r - r_0; n\rangle = \left\langle \begin{bmatrix}x - x_0\\y - y_0\end{bmatrix}; \begin{bmatrix}a_1 \\ -a_1\end{bmatrix}\right\rangle = \left\langle\begin{bmatrix}x - x_0 \\ y - y_0\end{bmatrix}; \begin{bmatrix}A \\ B\end{bmatrix} \right\rangle $$ What about line in space? Parametric equation doesn’t change, one vectors will get longer. Linear equation will change: now it’s not one equation, it’s system of two equations: $$ \begin{cases} a_2x - a_1y + (a_1y_0 - a_2x_0) = 0 \\ a_3y - a_2z + (a_2z_0 - a_3y_0) = 0 \end{cases} $$ Method with normal and translation not working in space!

Addition: let’s take to look at distance from point to line:

And we can detect where point relative to a line: group of points from one side, for which $Ax +By + C > 0$ is true; group from another side, for which $Ax +By + C < 0$ is true.

# The plane in space

Again this question. How to define a plane in space? Of course, through with three points non-collinear, that do not lie on the same straight line. Collinear points lie on the same line. And again let’s consider three methods:

1. Parametric equation: $$ \pi = \{r: r = r_0 + t_1v_1 + t_2v_2, \quad t_1, t_2 \in \mathbb{R}\} $$
2. Normal and translation is the same as for line: $\langle r - r_0; n\rangle = 0$
3. Linear equation: start from normal and translation and expand it: $$ \langle r - r_0; n\rangle = 0 \quad \Rightarrow \quad \underbrace{n_1}_{A} x\ + \ \underbrace{n_2}_{B} y\ + \ \underbrace{n_3}_{C} z \ \underbrace{- \langle r_0 ;n\rangle}_{D} = 0. \ \Rightarrow \ Ax + By + Cz + D = 0 $$ Let’s look to relationship between this methods:

• For rewrite parametric equation to normal and translation we need to find normal vector from this system of equations: $\begin{cases}\langle n; v_1\rangle = 0 \\ \langle n; v_2\rangle = 0\end{cases}$
• For rewrite normal and translation form to linear equation we can use this: $A = n_1$, $B = n_2$, $C = n_3$, $D = -\langle n ;r_0\rangle$
• If we have three points, defined plane, we can go to parametric equation: $\begin{array}a v_1 = r_1 - r_0\\ v_2 = r_2 - r_0\end{array}$

Projection point $P$ on plan $\pi$ we call point $P_\pi \in \pi$, the closest to the point $P$: $P_\pi = \arg{\min{||P - X||}}, \ X \in \pi$. Then, the distance from point $P$ to plane $\pi$ will be $d(P, \pi)$ — the distance between point $P$ and her projection $P_\pi$:

Detecting where point relative a plane is the same as line.