AHL 3.14—Vector equation of line

In coordinate geometry, we typically describe lines using equations like $y=mx+c$. But in three-dimensional space, this approach breaks down — a single scalar equation can't fully describe a line in 3D. The vector equation of a line solves this elegantly by using vectors to pin down both a point on the line and the direction it travels.

The vector equation of a line is:

$\mathbf{r}=\mathbf{a}+\lambda \mathbf{b}$

Where:

• $\mathbf{r}$ is the position vector of any point on the line
• $\mathbf{a}$ is the position vector of a known point on the line
• $\mathbf{b}$ is the direction vector (parallel to the line)
• $\lambda \in \mathbb{R}$ is a scalar parameter