Question

Given a point ( x, y
), what is the new point when the point is rotated counterclockwise about the origin 270 degrees?(

Answer 1

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 \((y, -x)\).