AHL 3.17—Vector equations of a plane

A plane in three-dimensional space is a flat, infinite two-dimensional surface. Unlike a line, which requires only one direction to describe, a plane requires two independent directions , meaning we need more information to pin it down uniquely.

There are three equivalent ways to represent a plane mathematically, each useful in different contexts:

1. Parametric (vector) form: $\mathbf{r}=\mathbf{a}+\lambda \mathbf{b}+\mu \mathbf{c}$
2. Normal (dot product) form: $(\mathbf{r}-\mathbf{a})\cdot \mathbf{n}=0$, equivalently $\mathbf{r}\cdot \mathbf{n}=\mathbf{a}\cdot \mathbf{n}$
3. Cartesian form: $ax+by+cz=d$

Being fluent in all three forms , and knowing how to convert between them , is essential for IB HL Mathematics.

The parametric vector equation of a plane is:

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

Parametric Equation of a Plane: An equation of the form $\mathbf{r}=\mathbf{a}+\lambda \mathbf{b}+\mu \mathbf{c}$, where $\mathbf{a}$ is a fixed point on the plane, $\mathbf{b}$ and $\mathbf{c}$ are two non-parallel vectors lying in the plane, and $\lambda ,\mu \in \mathbb{R}$ are free scalar parameters.

What each component does:

• $\mathbf{a}$ , anchors the plane at a known point in space
• $\mathbf{b}$ and $\mathbf{c}$ , define two independent directions along which the plane extends
• $\lambda$ and $\mu$ , scalar parameters that, as they range over all real values, sweep out the entire plane

Comparison with the line equation:

A line is one-dimensional in space; a plane is two-dimensional in space.