In a three dimensional space, consider the point $\displaystyle (x,y,z)=(4,3,1)$. Rotate this point 45 degrees counter-clockwise about the z-axis (as viewed from above). What are the coordinates of the new point?
Hi, Mrdavid445.
Since the rotation is about the z-axis, we are rotating the point (4,3,1) in the plane z=1. Since the rotation occurs in a plane, we can use the two-dimensional counterclockwise rotation matrix
$\displaystyle R=\begin{pmatrix}
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta
\end{pmatrix}$,
where $\displaystyle \theta$ is the angle of rotation - see Rotation matrix - Wikipedia, the free encyclopedia. If we multiply the vector $\displaystyle [4,3]^{T}$ (T is the transpose) times the rotation matrix (after plugging in a value for $\displaystyle \theta$) we will obtain the new x and y coordinates for the rotated point. Since the rotation is in the plane z=1, the z coordinate will still be 1.
Does that answer your question?
Good luck!