Question

In the 3D Cartesian system the rotation matrix is around the z-axis is (a 2D rotation):

cos()-sin(e) 0 Rsin() cos() 0 0
Where is the angle to rotate. Then rotation from A to A' is can be represented via matrix multiplications: [A'] = [R][A]

Such a rotation is useful to return a system solved in simplified co-ordinates to it's original co-ordinate system, returning to original meaning to the answer. A full 3D rotation is simply a series of 2D rotations (with the appropriate matrices)

Question: If

is the vector A = 1 x^ + 0y^ + 0z^, and B = 1x^ + 3y^ + 1z^ rotate both vectores through 45 and 90 degrees about the z-axis. Hint: Do A first, the answer should be logical. Note how the z^ component never changes, thus we have a 2D rotation around the z-axis.

Answer 1

For vector A(rotation by 45 degrees)

$\begin{pmatrix} \cos 45^{\degree} &-\sin 45^{\degree} &0 \\ sin 45^{\degree} &\cos 45^{\degree} &0 \\ 0 &0 &1 \end{pmatrix}\begin{pmatrix} 1\\0 \\0 \end{pmatrix}=\begin{pmatrix} \cos 45^{\degree}\\\sin 45^{\degree} \\0 \end{pmatrix}=\begin{pmatrix} \frac {1}{\sqrt {2}}\\\frac {1}{\sqrt {2}} \\0 \end{pmatrix}$

For vector B(rotation by 90 degrees)

$\begin{pmatrix} \cos 90^{\degree} &-\sin 90^{\degree} &0 \\ sin 90^{\degree}&\cos 90^{\degree} &0 \\ 0 &0 &1 \end{pmatrix}\begin{pmatrix} 1\\3 \\1 \end{pmatrix}=\begin{pmatrix} -3\\1 \\1 \end{pmatrix}$