A specific example of an orthogonal matrix

To answer flie first question, we write explicitly. 

Taking the dot products of each with each yields 

If we want instead to rotate the coordinate system without changing the vectors, we apply the inverse rotation 0. To derive the elements of we have merely to swap to obtain 

As a rotation transforms one orthogonal basis into another, a matrix satisfying (3.8) is said to be orthogonal. Note that matrices that perform improper rotations, i.e. that combine a rotation with a mirror reflection, also map orthogonal bases into other ones, and also satisfy 

We now consider a specific 0 that performs a counter-clockwise rotation (in the 12 plane) 

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