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Rotating the Plane


  You will need to enable Java in order to use this demonstration.  

Please be patient while the applet is loaded (60K)...

Interaction Controls

To rotate the coordinate axes around the origin, drag the mouse vertically over the applet while the Option or Alt key is held down.

The readout at the bottom represents the rotation matrix for the coordinate axes (see below).

When you are done with the demonstration, click on the up button at the top of this page, or use your browser's back button to go back to the previous page.

This demonstration is one of a series: you can view the 3D version, or move on to the 4D version next.

What this Demonstrates:

Dragging the mouse will cause the axes in two-space to rotate. The first column of the array at the bottom of the picture keeps track of the two new coordinates of the unit vector that points along the x axis. Similarly the second column represents the new position of the unit vector along the y axis.

You can add the squares of the components in each column to see that the length of either coordinate vector remains the same, i.e. equal to one, throughout the rotation.

Also, for those familiar with the dot product, it is possible to check that the columns of this array, or matrix as it is known, are perpendicular throughout. Any matrix whose columns form mutually perpendicular vectors is called an orthogonal matrix.

Once we know the positions of the two axes, we can apply the same rotation to any figure at all. In this case, we rotate a square along with the axes. This is the basis of the rotations that show up in computer animation, and in other applications in two-dimensional computer graphics.

Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.