Equation of a line in 3D

The previous articles dealt with describing a line in the plane. In this article we will show how to describe a line in 3D.

Parametric equations of a line in 3D

When we looked at the parametric equations of a line in the plane, we explained that if a line p passes through a point A and has a direction vector $\overrightarrow{\mathbf{u}}$, then every point X on the line p can be written as

$$X=A+t\cdot \overrightarrow{\mathbf{u}},t\in \mathbb{R}.$$

We call the number t the parameter. The equation above leads to the familiar system of equations

$$\begin{array}{rcl}x& =& a_1+t\cdot u_1\\ y& =& a_2+t\cdot u_2\end{array}$$

Let’s look at the figure:

If we add a scalar multiple of the direction vector (t times $\overrightarrow{\mathbf{u}}$) to the point A, we get the point B. And the same works for every point on the line.

This idea works just as well in 3D. In 3D we can also determine a line p by a point A that the line passes through and a direction vector $\overrightarrow{\mathbf{u}}$, and then every point X on the line satisfies

$$X=A+t\cdot \overrightarrow{\mathbf{u}},t\in \mathbb{R}.$$

We get exactly the same equation. The only difference is that we are in 3D, so the points X and A have three coordinates rather than two, and the direction vector $\overrightarrow{\mathbf{u}}$ has three components. The equation then gives a system of three equations instead of two:

$$\begin{array}{rcl}x& =& a_1+t\cdot u_1,\\ y& =& a_2+t\cdot u_2,\\ z& =& a_3+t\cdot u_3,\end{array}$$

where x, y, z are the coordinates of a point X(x, y, z) somewhere on the line, $\overrightarrow{\mathbf{u}}=(u_1,u_2,u_3)$ is the direction vector and A(a₁, a₂, a₃) is a point in 3D that the line passes through.