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 yaxis 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 rightangled triangle 

The xposition can be calculated from the radius r and the angle θ according to the following formula: 

The yposition 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 3space, counter clockwise through the angle φ about the zaxis shows the matrix below 


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