# Three Dimensional Space

• Aug 9th 2012, 04:30 PM
Mrdavid445
Three Dimensional Space
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?
• Aug 9th 2012, 07:24 PM
GJA
Re: Three Dimensional Space
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.