We all live in a three-dimensional world.
Everything we see or touch has length, depth, and width.
Most of the mathematics you have studied is either one- or two-dimensional. For instance, lines in a rectangular coordinate system have equations involving two variables, each with its own coordinate axis, where locations are specified by means of ordered pairs.
In three dimensions, three variables are needed. Each variable still has its own coordinate axis, and ordered triples are used to specify locations.
A linear equation in three variables is represented by a plane instead of a line in a three-dimensional coordinate system.
If two such planes intersect, they form a line.
If three planes intersect, they meet at a point.
This app will help you investigate the ways in which planes can interact in a three-dimensional coordinate system.
You enter the coefficients of the equations of up to three planes, including the constant in the equation. Individual equations may be toggled by using the checkbox. Selected equations are displayed as planes in the graphing area.
The options across the top control whether various features are included as part of the graph.
The “Show intersection” area in the lower right will toggle display of the lines of intersection of the specified pair of planes.
The point of intersection, if there is one, is plotted when the box in the lower left is checked.
The graphical output may be rotated for a better view by dragging the mouse across the screen.
Limited zooming is available by means of the up and down arrow keys or by using the mouse wheel.
Try all the options to see which help you understand the ideas as you consider the practice questions. Good Luck!