Draw the orthogonal axes for , and the rotated axes of by .
The equations are obtained by simple trig and vector addition.
A drawing normally explains this very easily, but since I suck at drawing (and cannot seem to find a picture online for this common transformation), I will try to explain it analytically.
Any axis can be defined by a unit vector parallel to it in the postive direction.
So let be the unit vectors parallel to the corresponding axes.
A point in the coordinate system can be represented by the vector .
For any vector , its representation in the (orthogonal) coordinate system is:
So the coordinate is given by .
So for , this is .
Similarly for ...
[Edit: Found an image here: http://www.tutornext.com/system/file...Fig.1.36_0.GIF,
perhaps this will help with intuition using simple trigonometry]