To rotate a point \((x, y)\) counterclockwise about the origin by an angle of 270 degrees, you can use the rotation formula. The rotation of a point \((x, y)\) by an angle \(\theta\) is given by:
\[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} \]
For a 270-degree rotation (which is equivalent to -90 degrees), we have:
\[ \cos(270^\circ) = 0 \quad \text{and} \quad \sin(270^\circ) = -1 \]
Substituting these values into the rotation formula gives us:
\[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} \]
Calculating the new coordinates:
\[ x' = 0 \cdot x + 1 \cdot y = y \] \[ y' = -1 \cdot x + 0 \cdot y = -x \]
Thus, the new coordinates after a 270-degree counterclockwise rotation are:
\[ (x', y') = (y, -x) \]
So, the new point after rotating the point \((x, y)\) counterclockwise about the origin by 270 degrees is \((y, -x)\).