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