How Do You Write the Vector Equation of a Plane?

A plane is written in vector form as $\mathbf{r}=\mathbf{a}+\lambda\mathbf{b}+\mu\mathbf{c}$, where $\mathbf{a}$ is the position vector of a point on the plane and $\mathbf{b}$ and $\mathbf{c}$ are two non-parallel vectors lying within it. This is IB Mathematics AA HL content from AHL 3.17: Vector equations of a plane.

The Method

In the vector equation of a plane,

$$\mathbf{r}=\mathbf{a}+\lambda\mathbf{b}+\mu\mathbf{c}, \qquad \lambda,\mu\in\mathbb{R}.$$

Changing $\lambda$ and $\mu$ generates every point on the plane. The two direction vectors must not be parallel; otherwise, the equation generates only a line.

If three non-collinear points $A$, $B$, and $C$ are given:

1. Use one point, usually $A$, to obtain $\mathbf{a}=\overrightarrow{OA}$.
2. Form $\mathbf{b}=\overrightarrow{AB}=\overrightarrow{OB}-$ and .

Why This Works

Starting from the fixed point represented by $\mathbf{a}$, the term $\lambda\mathbf{b}$ moves in one direction within the plane, while $\mu\mathbf{c}$ moves in a second direction. Because the vectors are non-parallel, their linear combinations cover two dimensions. You can confirm this by checking that $\mathbf{b}\times\mathbf{c}\neq\mathbf{0}$. Different choices of base point or direction vectors may produce different-looking but equivalent equations.

For example, let $A(1,2,0)$, $B(3,1,4)$, and $C(0,5,2)$. Then

$$\overrightarrow{AB}=\begin{pmatrix}2\\-1\\4\end{pmatrix},\qquad \overrightarrow{AC}=\begin{pmatrix}-1\\3\\2\end{pmatrix}.$$

Therefore,

$$\mathbf{r}=\begin{pmatrix}1\\2\\0\end{pmatrix}+\lambda\begin{pmatrix}2\\-1\\4\end{pmatrix}+\mu\begin{pmatrix}-1\\3\\2\end{pmatrix}.$$

Substituting $(\lambda,\mu)=(0,0),(1,0),(0,1)$ recovers $A$, $$, and , verifying that all three points lie on the plane.

Exam Technique

This method is valid for AA HL Paper 1 and Paper 2. Show both vector subtractions before stating the equation. A common misconception is to use the position vectors of $B$ and $C$ directly as direction vectors; instead, subtract the chosen base point.

On Paper 1, keep vectors exact and show subtraction component by component. On Paper 2, technology may check the cross product, but it does not replace required setup. If asked to verify that a point lies on the plane, substitute its coordinates and solve for a consistent pair of parameters. If no pair exists, the point is not on the plane.