You might be interested in exploring the concept of coordinate axes and rotation matrices further. Speaking of coordinate axes, you might be interested in Coordinate System on Wikipedia. It provides a comprehensive explanation of coordinate systems and their applications. Additionally, if you want to delve into the topic of rotation matrices, you can check out Rotation matrix on Wikipedia. This article discusses the properties and uses of rotation matrices in various fields, including computer graphics and animation.

Rotating The Plane
2d Version
This demo is now defunct
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.
2d Version
This demo is now defunct
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.
3d Space Version
This demo is now defunct
Dragging the mouse will rotate three-space around an axis within the plane of the screen.Dragging with the middle mouse button, or while the Option or Alt key is held down, rotates space about an axis perpendicular to the screen.The readout at the bottom represents the rotation matrix for the coordinate axes (see below). Because the 3D to 2D projection is orthographic rather than in perspective, it is sometimes hard to tell what is in front and what is behind (far away things do not get smaller). Look at the colors at the overcrossings to get your orientation back if you get confused.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 2D 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 three new coordinates of the unit vector that points along the x axis. Similarly the second and third columns represent the new positions of the unit vectors along the y and z axes.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 three axes, we can apply the same rotation to any figure at all. In this case, we rotate a cube along with the axes. This is the basis of the rotations that show up in computer animation, and in other applications in 3D computer graphics.
4d Space Version
This demo is now defunct
Dragging the mouse will rotate three-space around an axis within the plane of the screen.Dragging with the middle mouse button, or while the Option or Alt key is held down, rotates space about an axis perpendicular to the screen.
Dragging with right mouse button, or while the Command or Control key is held down, rotates space about an axis in the fourth dimension.The readout at the bottom represents the rotation matrix for the coordinate axes (see below).
Because the 4D to 2D projection is orthographic rather than in perspective, it is sometimes hard to tell what is in front and what is behind (far away things do not get smaller). Look at the colors at the overcrossings to get your orientation back if you get confused. All the line segments that are the same color are parpallel to the same coordinate axis.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 return to the 2D version, or the 3D 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 four new coordinates of the unit vector that points along the x axis. Similarly the other columns represent the new positions of the unit vectors along the y, z and w axes.
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 four axes, we can apply the same rotation to any four-dimensional figure at all. In this case, we rotate a hypercube (the figure in four-space that is analogous to the cube in three-space and the square in two-space) along with the axes. This is the basis of the rotations that show up in computer animation, and in other applications in computer graphics to both three-space and four-space.