Analytic Geometry: Lines

Lines

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:

\(ax+by+c=0\)

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:

\(y=mx+b\)

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.

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:

\(m=\frac{Δy}{Δx}=\frac{y_2-y_1}{x_2-x_1}\)

Notes:

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

\(α=tan^{-1}\big[\frac{m_2-m_1}{1+m_2m_1}\big]\)

Two lines are parallel if:

\(m_1=m_2\)

Two lines are perpendicular if:

\(m_1=-\frac{1}{m_2}\)

Intercept Form of a Line

A line with an equation of the form:

\(\frac{x}{a}+\frac{y}{b}=1\)

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:

\(\frac{x}{3}+\frac{y}{2}=1\)

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:

\(y-y_1=m(x-x_1)\)

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

\(y-4=2(x-5)\)

Which can be rearranged to give:

\(y=2x-6\)

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:

\(\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}\)

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:

\(-4y=x-5\)or\(y=-\frac{1}{4}x+20\)

Lines in Space

General Form of a Spacial Line

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

\(Ax+By+Cz+D=0\)

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 \((x_1,y_1,z_1)\):

\(\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}\)

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:

\(\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}\)

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 \((x_1,y_1,z_1)\).

\(x=x_1+ld\)
\(y=y_1+md\)
\(z=z_1+nd\)