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

The first line
First line scope (a₁)
Free parameter of the first line (b₁)
The second line
Second line scope (a₂)
Free parameter of the second line (b₂)
Others
Prefered angle measure unit

Results - what we can say about 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 $0.7086262721276703 \left[rad\right]$

Functions graph

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.

Tags and links to this website

Tags:

line_pair · two_lines_intersection · point_of_lines_intersection · pair_of_lines · straight_line_pair · straight_perpendicular · straight_parallel · two_lines_cross · are_lines_parallel · are_lines_perpendicular · parallel_lines_condition · perpendicular_lines_condition

Tags to Polish version:

wzajemne_polozenie_prostych · para_prostych · dwie_linie · linie_rownolegle · linie_prostopadle · punkt_przeciecia_dwoch_prostych · warunek_rownoleglosci_prostych · warunek_prostopadlosci_prostych

What tags this calculator has

CALCULATOR · MATH · A -> Z (all)

Permalink

This is permalink. Permalink is the link containing your input data. Just copy it and share your work with friends:

https://calculla.com/line_pair?inLineScope1=$symbolic(2)&inLineScope2=$symbolic(-4)&inFreeParameter1=$symbolic(3)&inFreeParameter2=$symbolic(5)&inAngleUnitId=rad

Links to external sites (leaving Calculla?)