Rotation of object relative to FIXED axis:
Basic equations you can get by looking at the diagram above:
$x_{1} = r \cos {\alpha} $ $x_{2} = r \cos {(\theta + \alpha)} $
$ y_{1} = r \sin {\alpha} $ $y_{2} = r \sin {(\theta + \alpha)}$
Using the equations above:
$\begin{align} x_{2} &= r \cos {(\theta + \alpha)} \\
&= r\cos \theta \cos \alpha – r\sin \theta \sin \alpha \\
&= (r\cos \alpha) \cos \theta – (r \sin \alpha) \sin \theta \\
&= x_{1} \cos \theta – y_{1} \sin \theta
\end{align}$
$\begin{align} y_{2} &= r \sin {(\theta + \alpha)} \\
&= r\sin \theta \cos \alpha + r\cos \theta \sin \alpha \\
&= (r\cos \alpha) \cos \theta + (r \sin \alpha) \cos \theta \\
&= x_{1} \sin \theta + y_{1} \cos \theta
\end{align}$
Hence, For an anti-clockwise rotation,
$ \begin{pmatrix}
x_{2} \\
y_{2}
\end{pmatrix} = \begin{pmatrix}
\cos \theta & – \sin \theta \\
\sin \theta & \cos \theta
\end{pmatrix}\begin{pmatrix}
x_{1} \\
y_{1}
\end{pmatrix}$
$\begin{pmatrix}
\cos \theta & – \sin \theta \\
\sin \theta & \cos \theta
\end{pmatrix}$ is called the rotation matrix. Its determinant is 1.
To find the clockwise rotation matrix, you can do the calculations again. OR you can just transpose the above matrix.
Hence, the clockwise rotation matrix is: $\begin{pmatrix}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{pmatrix}$
Rotation of coordinate axes:
Basic equations:
$ x’ = r \cos \alpha $ $x = r \cos (\theta + \alpha)$
$y’ = r \sin \alpha $ $y = r \sin (\theta + \alpha)$
Using the basic equations:
Equation 1: $\begin{align}x &= r\cos (\theta + \alpha) \\
&= r\cos \theta \cos \alpha – r \sin \theta \sin \alpha \\
&= x’ \cos \theta – y’ \sin \theta \end{align}$
Equation 2: $\begin{align}y &= r\sin (\theta + \alpha) \\
&= r\sin \theta \cos \alpha + r \cos \theta \sin \alpha \\
&= x’ \sin \theta + y’ \cos \theta \end{align}$
From equation 2,
$x’ = \frac{y – y’ \cos \theta}{\sin \theta}$ $y’ = \frac{y – x’ \sin \theta}{\cos \theta}$
Substitute the above 2 equations into equation 1 and you will get:
$y’ = -x \sin \theta + y\cos \theta$ $x’ = x \cos \theta + y \sin \theta$
Hence,
$ \begin{pmatrix}
x \\
y
\end{pmatrix} = \begin{pmatrix}
\cos \theta & – \sin \theta \\
\sin \theta & \cos \theta
\end{pmatrix}\begin{pmatrix}
x’ \\
y’
\end{pmatrix}$
$ \begin{pmatrix}
x’ \\
y’
\end{pmatrix} = \begin{pmatrix}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{pmatrix}\begin{pmatrix}
x \\
y
\end{pmatrix}$
Back To Mathematics For An Undergraduate Physics Course
