nD Rotation Matrices

  It's possible to find a matrix that will rotate a vector in a plane a given angle in n-space. To show this in 4D we choose two arbitrary unit vectors, a and b, to define the plane and make the second perpendicular to the first. Next we pick a unit vector, x, in the plane obtained by rotating the vector a by α = 5 degrees. The vector y is perpendicular to x in the ab-plane. We can find another vector, x', by rotating x and angle β in the plane using the formula found previously. The matrix R(α) shown below performs the same rotation as shown by the result x''.

If we do this again for α = 13 degrees finding a new vectors x, x' and x'' using the same method.

The matrix used will rotate any vector in the plane 10 degrees in the same direction.