Question

What is the mathematical transformation equivalent to rotating an object 90 degrees clockwise?

Answer 1

To rotate a point in a 2D Cartesian coordinate system 90 degrees clockwise around the origin, you can use a mathematical transformation represented by a rotation matrix. The rotation matrix for a clockwise rotation by an angle \(\theta\) is given by:

\[
R = \begin{bmatrix}
\cos(\theta) & \sin(\theta) \\
-\sin(\theta) & \cos(\theta)
\end{bmatrix}
\]

For a 90 degrees clockwise rotation, \(\theta = 90^\circ\) or \(\theta = \frac{\pi}{2}\) radians. The cosine and sine of 90 degrees are:

\[
\cos(90^\circ) = 0 \quad \text{and} \quad \sin(90^\circ) = 1
\]

Thus, the rotation matrix becomes:

\[
R = \begin{bmatrix}
0 & 1 \\
-1 & 0
\end{bmatrix}
\]

To rotate a point \( (x, y) \) by 90 degrees clockwise, you can apply this matrix to the coordinate vector:

\[
\begin{bmatrix}
x' \\
y'
\end{bmatrix} = R \cdot \begin{bmatrix}
x \\
y
\end{bmatrix} = \begin{bmatrix}
0 & 1 \\
-1 & 0
\end{bmatrix} \cdot \begin{bmatrix}
x \\
y
\end{bmatrix}
\]

Carrying out the matrix multiplication, you get:

\[
\begin{bmatrix}
x' \\
y'
\end{bmatrix} = \begin{bmatrix}
0 \cdot x + 1 \cdot y \\
-1 \cdot x + 0 \cdot y
\end{bmatrix} = \begin{bmatrix}
y \\
-x
\end{bmatrix}
\]

Therefore, the new coordinates after a 90-degree clockwise rotation of the point \( (x, y) \) will be:

\[
(x', y') = (y, -x)
\]