Analytics geometry: point of lines intersection calculator
Calculator shows how your straight lines pair are positioned on one plane. It draws a chart of given straight lines if possible. Also, it calculates an angle between those lines.

# Beta version

BETA TEST VERSION OF THIS ITEM
This online calculator is currently under heavy development. It may or it may NOT work correctly.
You CAN try to use it. You CAN even get the proper results.
However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.
Feel free to send any ideas and comments !

# Calculations data - line scope and free coefficient of your lines

 Summary Formula of the first function Show source$y=2~ x+3$ Formula of the second function Show source$y=-4~ x+5$ Lines cross point Point of intersection of lines Show source$\left(\frac{1}{3},\frac{11}{3}\right)$ Lines position Are lines parallel no ✗ Are lines perpendicular no ✗ Angle between lines Tangent of angle Show source$\frac{6}{7}$ Angle Show source$0.7086262721276703 \left[rad\right]$

# Some facts

• Two lines k and l are parallel if they have identical line scope:
\begin{aligned} k: y & = \boxed {a} ~ x + b_1 \\ l: y & = \boxed {a} ~ x + b_2 \end{aligned}
where:
• a - line scope, the same for both straight lines,
• b1 - free coefficient of the first straight line,
• b2 - free coefficient of the second straight line.
• Two lines k and l are perpendicular if line scope of the first is negative inverse of the second one:
$a_1 = - \dfrac{1}{a_2}$
\begin{aligned} k: y & = \boxed{a} ~ x + b_1 \\ l: y & = \boxed{- \dfrac{1}{a}} ~ x + b_2 \end{aligned}
where:
• a - scope of the first line,
• b1 - free coefficient of the first straight line,
• b2 - free coefficient of the second straight line.
• If lines are parallel, then they have no common point. Colloquially, one never crosses another one.
• If lines are not parallel, then they intersect exactly once in the point with below coordinates:
$\begin{cases} P_x = \dfrac {b_2 - b_1}{a_1 - a_2} \\ P_y = a_1 ~ \dfrac{b_2 - b_1}{a_1 - a_2} + b_1 \end{cases}$
where:
• a1 - scope of the first line,
• a2 - scope of the second line,
• b1 - free coefficient of the first straight line,
• b2 - free coefficient of the second straight line.
• To calculate the measure of angle created between straight lines we can use the following formula:
$\alpha = \arctan\left(\left| \dfrac{a_2 - a_1}{1 + a1 ~a2} \right| \right)$
where:
• $\alpha$ - angle measure between lines,
• $a_1$ - scope of the first line,
• $a_2$ - scope of the second line.
• If you want to know more about straight line, its formula and properties, check our other calculator: Straight line.