Line in plane
We found 18 formulas for line in plane
1.
Normal vector to a line
The vector normal to the line $Ax+By+C=0$ is $\vec{n}(A,B)$.
4.
Equation of a line given point and the gradient.
$ y = y_0 + k \left(x-x_0\right) $
5.
Line equation trought two points - version 1
$ \dfrac{y-y_1}{y-y_2} = \dfrac{x-x_1}{x-x_2}$
6.
Line equation throught two points
$\begin{vmatrix} x&y&1\\x_1&y_1&1 \\ x_2&y_2&1 \end{vmatrix} = 0$
9.
Point - direction form of a line equation
$\dfrac{x-x_1}{X} = \dfrac{y-y_1}{Y}$
10.
Line in prametric form
$\begin{aligned}
x &= a_1+tb_1 \\[0.3 em]
y &= a_2+tb_2
\end{aligned}$
11.
Line - point distance
The distance from the point $P(x_0,y_0)$ to the line $Ax+By+C=0$ is
$d=\dfrac{|Ax_0 + By_0 + C|}{\sqrt{A^2+B^2}}$
12.
Parallel lines - verison 1
Lines $y=k_1x + b_1$ and $y=k_2x + b_2$ are parralel iff $k_1 = k_2$.
13.
Parallel lines - verison 2
Lines $A_1x+B_1y+C_1=0$ and $A_2x+B_2y+C_2=0$ are parallel iff $\dfrac{A_1}{A_2} = \dfrac{B_1}{B_2}$.
14.
Perpendicular lines - verison 1
Lines $y=k_1x + b_1$ and $y=k_2x+b_2$ are perpendicular iff $k_1\cdot k_2=1$.
15.
Perpendicular lines - verison 2
Lines $A_1x+B_1y+C=0$ and $A_2x+B_2y+C=0$ are perpendicular iff $A_1A_2+B_1B_2=0$.
16.
Angle between two lines - verison 1
$\tan\phi = \dfrac{k_2-k-1}{1+k_1k_2}$
17.
Angle between two lines - version 2
$\cos\phi = \dfrac{A_1A_2+B_1B_2}{\sqrt{A_1^2+B_1^2} \cdot \sqrt{A_2^2+B_2^2}}$
18.
Intersection of two lines
Intersection point of lines $A_1x+B_1y+C_1=0$ and $A_2x+B_2y+C_2=0$ has coordinates
$
x_0 = \dfrac{B_1C_2 - B_2C_1}{A_1B_2-A_2B_1},
y_0 = \dfrac{-A_1C_2 + A_2C_1}{A_1B_2-A_2B_1}
$