Search

Search results:

1.
Normal vector to a line
The vector normal to the line Ax+By+C=0Ax+By+C=0 is n(A,B)\vec{n}(A,B).
2.
Slope-intercept form of straight line
y=kx+ny=kx+n
3.
Gradient of a line
k=tanα=y2y1x2x2k= \tan \alpha = \dfrac{y_2-y_1}{x_2-x_2}
4.
Equation of a line given point and the gradient.
y=y0+k(xx0) y = y_0 + k \left(x-x_0\right)
5.
Line equation trought two points - version 1
yy1yy2=xx1xx2 \dfrac{y-y_1}{y-y_2} = \dfrac{x-x_1}{x-x_2}
6.
Line equation throught two points
xy1x1y11x2y21=0\begin{vmatrix} x&y&1\\x_1&y_1&1 \\ x_2&y_2&1 \end{vmatrix} = 0
7.
Intercept form of the line equation
xa+yb=1\dfrac{x}{a} + \dfrac{y}{b} = 1
8.
Normal form of a line
xcosα+ysinα=px\cos\alpha + y\sin\alpha = p
9.
Point - direction form of a line equation
xx1X=yy1Y\dfrac{x-x_1}{X} = \dfrac{y-y_1}{Y}
10.
Line in prametric form
x=a1+tb1y=a2+tb2\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(x0,y0)P(x_0,y_0) to the line Ax+By+C=0Ax+By+C=0 is d=Ax0+By0+CA2+B2d=\dfrac{|Ax_0 + By_0 + C|}{\sqrt{A^2+B^2}}
12.
Parallel lines - verison 1
Lines y=k1x+b1y=k_1x + b_1 and y=k2x+b2y=k_2x + b_2 are parralel iff k1=k2k_1 = k_2.
13.
Parallel lines - verison 2
Lines A1x+B1y+C1=0A_1x+B_1y+C_1=0 and A2x+B2y+C2=0A_2x+B_2y+C_2=0 are parallel iff A1A2=B1B2\dfrac{A_1}{A_2} = \dfrac{B_1}{B_2}.
14.
Perpendicular lines - verison 1
Lines y=k1x+b1y=k_1x + b_1 and y=k2x+b2y=k_2x+b_2 are perpendicular iff k1k2=1k_1\cdot k_2=1.
15.
Perpendicular lines - verison 2
Lines A1x+B1y+C=0A_1x+B_1y+C=0 and A2x+B2y+C=0A_2x+B_2y+C=0 are perpendicular iff A1A2+B1B2=0A_1A_2+B_1B_2=0.
16.
Angle between two lines - verison 1
tanϕ=k2k11+k1k2\tan\phi = \dfrac{k_2-k-1}{1+k_1k_2}
17.
Angle between two lines - version 2
cosϕ=A1A2+B1B2A12+B12A22+B22\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 A1x+B1y+C1=0A_1x+B_1y+C_1=0 and A2x+B2y+C2=0A_2x+B_2y+C_2=0 has coordinates x0=B1C2B2C1A1B2A2B1,y0=A1C2+A2C1A1B2A2B1 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}