To rotate a point \((x, y)\) clockwise about the origin by 270 degrees, we can use the rotation transformation formulas.
A clockwise rotation of 270 degrees is equivalent to a counterclockwise rotation of 90 degrees. The transformation formulas for a counterclockwise rotation by an angle \(\theta\) are given by:
\[ \begin{pmatrix} x' \ y' \end{pmatrix}
\begin{pmatrix} x \cos \theta - y \sin \theta \ x \sin \theta + y \cos \theta \end{pmatrix} \]
For a 90-degree counterclockwise rotation, \(\theta = 90^\circ\) gives us:
\(\cos(90^\circ) = 0\)
\(\sin(90^\circ) = 1\)
Applying these values to the transformation:
\[ \begin{pmatrix} x' \ y' \end{pmatrix}
\begin{pmatrix} x \cdot 0 - y \cdot 1 \ x \cdot 1 + y \cdot 0 \end{pmatrix}
\begin{pmatrix} -y \ x \end{pmatrix} \]
Therefore, to obtain the new point after a 270-degree clockwise rotation, the new coordinates \((x', y')\) will be:
\[ (x', y') = (-y, x) \]
This means that the point \((x, y)\) after a 270-degree clockwise rotation about the origin becomes \((-y, x)\).