Multivariable calculus and vector analysis

Parametrization of a plane

A plane is determined by a point $\mathbf{p}$ in the plane and two vectors $\mathbf{a}$, and $\mathbf{b}$ parallel to the plane. (This is identical to the case with three points in the plane. Why?)

From this fact, we can parametrize a plane just like we parametrized a line (only we’ll need two parameters instead of one). If $\mathbf{x}$ is a point in the plane, the vector from $\mathbf{p}$ to $\mathbf{x}$ (i.e., $\mathbf{x}-\mathbf{p}$) is some multiple of $\mathbf{a}$ plus some multiple of $\mathbf{b}$. (Can you see why?) We can express this as $\mathbf{x}-\mathbf{p}=s\mathbf{a}+t\mathbf{b}$ for $t,s\in \mathbf{R}$. Usually, we’ll write this as $\mathbf{x}=\mathbf{p}+s\mathbf{a}+t\mathbf{b}$. The real numbers $s$ and $t$ are the parameters for this parametrization of the line plane.

The idea of the parametrization is that as the parameters $s$ and $t$ sweep through all real numbers, the point $\mathbf{x}$ sweeps out the plane. In other words, it is a two-dimensional analogue of the parametrization of the line.

Here we’ve added the point $\mathbf{x}$ in cyan. You can’t move $\mathbf{x}$ directly, but you can move it by changing the parameters $s$ and $t$ (the blue points on sliders).

Clearly, for any value of $s$ and $t$, the point $\mathbf{x}$ lies on the plane. Also, by changing $s$ and $t$, you can move the point $\mathbf{x}$ to any position on the plane.

Contents by topic

1. Geometry in higher dimensions

1a. Preliminaries

1b. Lines and planes

• Parametrization of a line
• Forming planes
• A line or a plane or a point?
• Parametrization of a plane
• Distance from point to plane
• Parametrization of a line examples
• Forming plane examples
• Plane parametrization example
• Distance from point to plane example
• Intersecting planes example

1c. Matrices

1d. Visualizing functions in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

5. The fundamental theorems of vector calculus

6. Review material