Question

Which of the following mapping is used for a 270 degree counterclockwise

Answer 1

To represent a mapping corresponding to a 270-degree counterclockwise rotation, especially in a 2D Cartesian coordinate system, you can use the following transformation.

For a point \((x, y)\), the transformation under a 270-degree counterclockwise rotation can be calculated using the rotation matrix:

\[
\begin{pmatrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{pmatrix}
\]

Where \(\theta\) is the angle of rotation in radians. For a 270-degree rotation, which is equivalent to \(-90\) degrees (or \(-\frac{\pi}{2}\) radians), the cosine and sine values are:

\[
\cos(-90^\circ) = 0, \quad \sin(-90^\circ) = -1
\]

So the rotation matrix becomes:

\[
\begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix}
\]

Now applying this matrix to the point \((x, y)\):

\[
\begin{pmatrix}
x' \\
y'
\end{pmatrix}
=
\begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
y \\
-x
\end{pmatrix}
\]

Thus, the mapping for a 270-degree counterclockwise rotation is:

\[
(x, y) \rightarrow (y, -x)
\]