IB Maths AA 3.12 Notes

Equations of a plane

A plane is a flat two-dimensional surface that extends infinitely in all directions. In three-dimensional geometry, planes can be described using vector equations or Cartesian equations. A plane can be written in vector form as:

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

where $\mathbf{a}$ is the position vector of a fixed point on the plane, and $\mathbf{b}$ and $\mathbf{c}$ are two non-parallel vectors that lie in the plane.

The parameters $\lambda$ and $\mu$ can take any real values. As they vary, $\mathbf{r}$ traces out every point on the plane.

Write the vector equation of a plane passing through the point $(1,2,3)$ and containing the vectors $\begin{pmatrix}2\\1\\0\end{pmatrix}$ and $\begin{pmatrix}1\\-1\\4\end{pmatrix}$.

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

A plane can also be described using a normal vector. A normal vector is perpendicular to the plane. If $\mathbf{n}$ is a normal vector and $\mathbf{a}$ is the position vector of a point on the plane, then the plane can be written as:

$\mathbf{r}\cdot\mathbf{n}=\mathbf{a}\cdot\mathbf{n}$

This works because every point $\mathbf{r}$ on the plane has the same component in the direction of the normal.

A plane has normal vector $\begin{pmatrix}2\\-1\\3\end{pmatrix}$ and passes through the point $(1,4,2)$. Determine its equation.

Then $\mathbf{a}\cdot\mathbf{n}=\begin{pmatrix}1\\4\\2\end{pmatrix}\cdot\begin{pmatrix}2\\-1\\3\end{pmatrix}=2-4+6=4$.

So the plane is $\mathbf{r}\cdot\begin{pmatrix}2\\-1\\3\end{pmatrix}=4$.

If the normal vector is $(a,b,c)$, then the Cartesian equation of the plane is:

$ax+by+cz=d$

Here, $(a,b,c)$ is the normal vector.

Convert the plane $\mathbf{r}\cdot\begin{pmatrix}2\\-1\\3\end{pmatrix}=4$ to Cartesian form.

If $\mathbf{r}=\begin{pmatrix}x\\y\\z\end{pmatrix}$, then $\begin{pmatrix}x\\y\\z\end{pmatrix}\cdot\begin{pmatrix}2\\-1\\3\end{pmatrix}=4$.

So $2x-y+3z=4$.