3D Geometry Formulae (Part 1)

This lesson collects the key formulae of three dimensional geometry: direction cosines and ratios, the equations of a line, the angle between two lines, and the distances involving skew and parallel lines.

The three axes and direction cosines

In three dimensions there are three mutually perpendicular axes: the $x$, $y$ and $z$ axes. A line in space makes angles $\alpha$, $\beta$, $\gamma$ with these axes. These are the direction angles, and their cosines are the direction cosines, written $l$, $m$, $n$:

$l = \cos\alpha, \qquad m = \cos\beta, \qquad n = \cos\gamma$

The direction cosines always satisfy

$l^2 + m^2 + n^2 = 1,$

which is the same as

$\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1.$

Direction ratios

Numbers proportional to the direction cosines are called direction ratios, written $a$, $b$, $c$. The direction cosines are recovered from them by dividing by the length:

$l = \pm\frac{a}{\sqrt{a^2 + b^2 + c^2}}, \qquad m = \pm\frac{b}{\sqrt{a^2 + b^2 + c^2}}, \qquad n = \pm\frac{c}{\sqrt{a^2 + b^2 + c^2}}.$

The sign depends on the octant, but in most cases we take the positive values.

Equation of a line through a point

For a line through the point $P(x_1, y_1, z_1)$ and parallel to a vector $\vec b$, there are two forms.

Vector form. With $\vec a$ the position vector of $P$,

$\vec r = \vec a + \lambda\,\vec b.$

Cartesian form. With $a$, $b$, $c$ the direction ratios of the parallel vector,

$\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}.$

Equation of a line through two points

For a line through $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$, with position vectors $\vec a$ and $\vec b$:

Vector form.

$\vec r = \vec a + \lambda\,(\vec b - \vec a).$

Cartesian form.

$\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}.$

Angle between two lines

Take two lines with direction vectors $\vec b_1$ and $\vec b_2$, or direction ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$.

Vector form.

$\cos\theta = \frac{\vec b_1 \cdot \vec b_2}{|\vec b_1|\,|\vec b_2|}.$

Cartesian form.

$\cos\theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2}\,\sqrt{a_2^2 + b_2^2 + c_2^2}}.$

Perpendicular and parallel conditions

Perpendicular lines. The lines meet at right angles when

$\vec b_1 \cdot \vec b_2 = 0, \qquad \text{or} \qquad a_1 a_2 + b_1 b_2 + c_1 c_2 = 0.$

Parallel lines. The direction vectors are proportional:

$\vec b_1 \times \vec b_2 = \vec 0, \qquad \text{or} \qquad \vec b_1 = \lambda\,\vec b_2,$

which in Cartesian form is

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}.$

Distance between skew and parallel lines

Skew lines. For $\vec r = \vec a_1 + \lambda\,\vec b_1$ and $\vec r = \vec a_2 + \mu\,\vec b_2$ (here $\vec b_1$ and $\vec b_2$ differ), the shortest distance is

$d = \frac{\bigl|(\vec b_1 \times \vec b_2) \cdot (\vec a_2 - \vec a_1)\bigr|}{|\vec b_1 \times \vec b_2|}.$

In Cartesian form this is

$d = \frac{\left| \begin{matrix} x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{matrix} \right|}{\sqrt{(a_1 b_2 - a_2 b_1)^2 + (b_1 c_2 - b_2 c_1)^2 + (c_1 a_2 - c_2 a_1)^2}}.$

Parallel lines. For $\vec r = \vec a_1 + \lambda\,\vec b$ and $\vec r = \vec a_2 + \mu\,\vec b$ (the same $\vec b$), the distance is

$d = \frac{\bigl|\vec b \times (\vec a_2 - \vec a_1)\bigr|}{|\vec b|}.$

Key takeaways

• Direction cosines satisfy $l^2 + m^2 + n^2 = 1$, and direction ratios are any numbers proportional to them.
• A line is fixed by a point and a direction, giving matching vector and Cartesian equations.
• Two lines are perpendicular when $\vec b_1 \cdot \vec b_2 = 0$ and parallel when $\vec b_1 \times \vec b_2 = \vec 0$.
• The skew line distance uses $\vec b_1 \times \vec b_2$, while the parallel line distance uses the single shared direction $\vec b$.