# 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 !

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

The first line | ||

First line scope (a_{1}) | ||

Free parameter of the first line (b_{1}) | ||

The second line | ||

Second line scope (a_{2}) | ||

Free parameter of the second line (b_{2}) | ||

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,

**b**- free coefficient of the first straight line,_{1}

**b**- free coefficient of the second straight line._{2}

- Two lines
**k**and**l**are**perpendicular**if**line scope**of the first is**negative inverse**of the second one:

$a_1 = - \frac{1}{a_2}$$\begin{aligned} k: y & = \boxed{a} ~ x + b_1 \\ l: y & = \boxed{- \frac{1}{a}} ~ x + b_2 \end{aligned}$where:

**a**- scope of the first line,

**b**- free coefficient of the first straight line,_{1}

**b**- free coefficient of the second straight line._{2}

- 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:

$\begin{cases} P_x = \frac {b_2 - b_1}{a_1 - a_2} \\ P_y = a_1 ~ \frac{b_2 - b_1}{a_1 - a_2} + b_1 \end{cases}$where:

**a**- scope of the first line,_{1}

**a**- scope of the second line,_{2}

**b**- free coefficient of the first straight line,_{1}

**b**- free coefficient of the second straight line._{2}

- To calculate the measure of angle created between straight lines we can use the following formula:

$\alpha = \arctan\left(\left| \frac{a_2 - a_1}{1 + a1 ~a2} \right| \right)$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:

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