Math Insight

Parametrization of a plane

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

Applet: Plane determined from three points

The Java applet did not load, and the above is only a static image representing one view of the applet. The applet was created with LiveGraphics3D. The applet is not loading because it looks like you do not have Java installed. You can click here to get Java.

Applet: Plane determined from three points

Plane determined from three points. The plane is determined by the points $\color{red}{P}$ (in red), $\color{green}{Q}$ (in green), and $\color{blue}{R}$ (in blue), which you can move by dragging with the mouse. The vectors from $\color{red}{P}$ to both $\color{green}{Q}$ and $\color{blue}{R}$ are drawn in the corresponding colors. The normal vector (in cyan) is the cross product of the green and blue vectors.

More information about applet.

From this fact, we can parametrize a plane just like we parametrized a line (only we'll need two parameters instead of one). If $\vc{x}$ is a point in the plane, the vector from $\vc{p}$ to $\vc{x}$ (i.e., $\vc{x}-\vc{p}$) is some multiple of $\vc{a}$ plus some multiple of $\vc{b}$. (Can you see why?) We can express this as $\vc{x}-\vc{p} = s\vc{a} + t\vc{b}$ for $t,s \in \vc{R}$. Usually, we'll write this as $\vc{x} = \vc{p} + s\vc{a} + t \vc{b}$. 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 $\vc{x}$ 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 $\vc{x}$ in black. You can't move $\vc{x}$ directly, but you can move it by changing the parameters $s$ and $t$ (the blue points on sliders).

Applet: Parametrization of a plane

The Java applet did not load, and the above is only a static image representing one view of the applet. The applet was created with LiveGraphics3D. The applet is not loading because it looks like you do not have Java installed. You can click here to get Java.

Applet: Parametrization of a plane

Parametrization of a plane. The plane is determined by the point $\color{red}{\vc{p}}$ (in red) and the vectors $\color{green}{\vc{a}}$ (in green) and $\color{blue}{\vc{b}}$ (in blue), which you can move by dragging with the mouse. The point $\vc{x} = \vc{p} + s\vc{a} + t \vc{b}$ (in black) sweeps out all points in the plane as the parameters $s$ and $t$ sweep through their values (which you can change by moving the points on the slider).

More information about applet.

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

You can see an example of parametrizing a plane here.