AHL 3.11—Vector equation of a line in 2d and 3d

One of the most powerful tools in analytical geometry is the vector equation of a line. Unlike the familiar $y=mx+c$ form, the vector equation works seamlessly in both 2D and 3D space — making it indispensable for IB DP Mathematics AI HL.

The vector equation of a line is written as:

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

where each component has a precise geometric meaning:

• $\mathbf{r}$ — the position vector of any point on the line
• $\mathbf{a}$ — the position vector of a known, fixed point on the line
• $\mathbf{b}$ — the direction vector, parallel to the line
• $\lambda$ — a scalar parameter that can take any real value

As $\lambda$ ranges over all real numbers ($\lambda \in \mathbb{R}$), the vector $\mathbf{r}$ traces out every point on the line — extending infinitely in both directions.