Line and plane

2025-06-28

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.