What are planes

In math, planes are not flying objects. They are flat surfaces on a 3D graph.

Most planes, as well as other 3D objects are described by a equation for $z$ in terms of $x$ and $y$:

$z = cx + ky + n$

Where $c$, $k$ and $n$ are constants.

However, they can also be described in other ways too.

Here's an example of a plane in three space:

Notice how there is two independent variables $x$ and $y$ instead of only $x$ to obtain $z$.

Vector equation of a plane

The vector equation of a plane is similar to that of a line, however, since it is a plane, we need two vectors instead of one to describe it.

$\vec {r} = \vec {r_0} + s \vec {a} + t \vec {b} \quad \newline {or} \newline (x, \ y, \ z) = (x_0, \ y_0, \ z_0) + s(a_1, \ a_2, \ a_3) + t(b_1, \ b_2, \ b_3)$

Where:

• $s, t \in \mathbb {R}$

• $\vec {r} = (x, \ y, \ z)$ is a position vector at any point on the plane.

• $\vec {r_0} = (x_0, \ y_0, \ z_0)$ is a position vector at some point on the plane.

• $\vec {a} = (a_1, \ a_2, \ a_3)$ is a direction vector parallel to the plane.

• $\vec {b} = (b_1, \ b_2, \ b_3)$ is another direction vector parallel to the plane.

Here's what all the vectors look like compared to the plane:

Parametric equation of a plane

Similar to the parametric equations of a line, you can also obtain the parametric equations of a plane:

$x = x_0 + s a_1 + t b_1 \newline y = y_0 + s a_2 + t b_2 \newline z = z_0 + s a_3 + t b_3$

Again, these equations are used to determine the separate coordinate values:

Cartesian equation of a plane

Planes also have a cartesion (or scalar) equation form. The coefficients on the variables also denote a normal (or orthogonal) vector to the plane.

$Ax + By + Cz + D = 0$

So, its non-zero normal vector would be $\vec {n} = (A, \ B, \ C)$.

Again, here's a video to demonstrate what it looks like: