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A line, by definition, is a curve that geometrically is determined by two of its points. Our discussion here is on straight (linear) lines. A linear line is a one-dimensional construct: it has length. As such, a linear line can exist in any coordinate system with one or more dimensions (we can probably interpret this to mean that a linear line can exist in any coordinate system, since a dimensionless coordinate system is non-existent.)

General Form of a Line

In plane geometry, a line is a set of points satisfying a linear equation of the form:


Where \(a\) and \(b\) are not both zero.

In Cartesian coordintes, the equation of a straight line may take various standard forms, all of which are special cases of the general form.

Standard Form/Slope-Intercept Form

A line with the equation:


has a gradient (slope) of \(m\) and an intercept of \(b\) on the y-axis. For instance, the line \(y=3x+5\) has a gradient of 3 (the angle between the line and the x-axis is \(tan^{-1}\,3\)), and it cuts the y-axis at point (0,5). Basically, this equation says that for every 3 units we move to the right along the x-axis, we move 5 units upward on the y-axis.

Slope Intercept
Slope Intercept

A plot of the line \(y=3x+5\) shows the intercepts at (-3,0) and (0,5).

The slope of the line is calculated as:



The angle between two lines with slopes \(m_1\) and \(m_2\) is:


Two lines are parallel if:


Two lines are perpendicular if:


Intercept Form of a Line

A line with an equation of the form:


intercepts the x-axis at \((a,0)\) and the y-axis at \((0,b)\). For example, the line \(3y=2x-6\) can be put in the form:


The intercept on the x-axis is 3 and the intercept on the y-axis is -2.

Point-Slope Form of a Line

A line with a slope \(m\) passing through a known point \((x_1,y_1)\) has the equation:


For example, a line with a gradient of 2 passing through the point (5,4) gives:


Which can be rearranged to give:


Two-Point Form of a Line

A line passing through two known points \((x_1,y_1)\) and \((x_2,y_2)\) has an equation of the form:


For example, a line passing through the points (3,2) and (-5,4) has the equation:

\(\frac{x-3}{-5-3}=\frac{y-2}{4-2}\) → \(\frac{x-3}{-8}=\frac{y-2}{2}\)

Which, after rearranging gives:


Lines in Space

General Form of a Spacial Line

The general form of a line in space is given by the equation:


The line can be written in many other forms, which are all variations of the general form.

Symmetrical Form a Spacial Line (Standard Form)

The equation is written in terms of direction numbers {\(l,m,n\)} and a point on the line \((x_1,y_1,z_1)\):


Two-Point Form of a Spacial Line

The equation is written in terms of two points on the line with coordinates \((x_1,y_1,z_1)\) and \((x_2,y_2,z_2)\). It has the form:


Parametric Form of a Spacial Line

The line is descibed in terms of its direction cosines \(\{l,m,n\}\), a point on the line \((x_1,y_1,z_1)\), and a variable parameter \(d\), which specifies the distance of the variable point \((x,y,z)\) from point \((x_1,y_1,z_1)\).


Analytic Geometry: Points Analytic Geometry: Line Intersection