# line in plane

## Equation of a line

Suppose $a,b,c\in \mathbb{R}$. Then the set of points $(x,y)$ in the plane that satisfy

$$ax+by+c=\mathrm{\hspace{0.33em}0},$$ |

where $a$ and $b$ can not be both 0, is an (infinite) *line*.

The value of $y$ when $x=0$, if it exists, is called the *$y$-intercept*. Geometrically, if $d$ is the $y$-intercept, then $(0,d)$ is the point of intersection of the line and the $y$-axis. The $y$-intercept exists iff the line is not parallel^{} to the $y$-axis. The *$x$-intercept* is defined similarly.

If $b\ne 0$, then the above equation of the line can be rewritten as

$$y=mx+d.$$ |

This is called the *slope-intercept form ^{}* of a line, because both the slope and the $y$-intercept are easily identifiable in the equation. The slope is $m$ and the $y$-intercept is $d$.

Three finite points $({x}_{1},{y}_{1})$, $({x}_{2},{y}_{2})$, $({x}_{3},{y}_{3})$ in ${\mathbb{R}}^{2}$ are collinear^{} if and only if the following determinant^{} vanishes:

$$\left|\begin{array}{ccc}\hfill {x}_{1}\hfill & \hfill {x}_{2}\hfill & \hfill {x}_{3}\hfill \\ \hfill {y}_{1}\hfill & \hfill {y}_{2}\hfill & \hfill {y}_{3}\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right|=0.$$ |

Therefore, the line through distinct points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$ has equation

$$\left|\begin{array}{ccc}\hfill {x}_{1}\hfill & \hfill {x}_{2}\hfill & \hfill x\hfill \\ \hfill {y}_{1}\hfill & \hfill {y}_{2}\hfill & \hfill y\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right|=0,$$ |

or more simply

$$({y}_{1}-{y}_{2})x+({x}_{2}-{x}_{1})y+{y}_{2}{x}_{1}-{x}_{2}{y}_{1}=0.$$ |

## Line segment

Let ${p}_{1}=({x}_{1},{y}_{1})$ and ${p}_{2}=({x}_{2},{y}_{2})$ be distinct points in ${\mathbb{R}}^{2}$. The closed line segement generated by these points is the set

$$\{p\in {\mathbb{R}}^{2}\mid p=t{p}_{1}+(1-t){p}_{2},\mathrm{\hspace{0.33em}0}\le t\le 1\}.$$ |

Title | line in plane |

Canonical name | LineInPlane |

Date of creation | 2013-03-22 15:18:29 |

Last modified on | 2013-03-22 15:18:29 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 17 |

Author | matte (1858) |

Entry type | Definition |

Classification | msc 53A04 |

Classification | msc 51N20 |

Synonym | y-intercept |

Synonym | x-intercept |

Related topic | LineSegment |

Related topic | SlopeAngle |

Related topic | LineInSpace |

Related topic | Slope |

Related topic | AnalyticGeometry |

Related topic | FanOfLines |

Related topic | PencilOfConics |

Defines | $y$-intercept |

Defines | $x$-intercept |

Defines | slope-intercept form |