RedCrab Math Tutorial

Matrices and Rotation

A polar coordinate can be described by a pair of numbers (x, y). The numbers are the distance of A from the y axis (x), and from the x axis (y) in the coordinate system. Any points to the left of the y-axis will have a negative x coordinate. Any points below the x axis, will have a negative y coordinate.

Instead with the term (x, y), we can describe the point with radius r and the angle θ (r, θ).

In the diagram above, r is the hypotenuse of a right-angled triangle

The x-position can be calculated from the radius r and the angle θ according to the following formula:
The y-position is calculated accordingly from the formula:

In the following figure, we have rotated the point (x, y) by the angle φ. So, we have now:

For the following trigonometric equation

we can write

and we become

When we write this in a matrix form it looks like this

The example below shows a matrix that rotates the vector by an angle of φ = 30 °.

With this Matrix the position vector for the point (1,0) becomes

Calculation and graphical representation in RedCrab Calculator

A rotation in 3-space, counter clockwise through the angle φ about the z-axis shows the matrix below