Question

1. what is the result of rotating a point (x, y) 90 degrees counterclockwise about the origin? (-y, x) (y, -x) (x, -y) (-x, -y)

Explanation:

Step1: Recall rotation formula

When rotating a point \((x,y)\) \(90^{\circ}\) counter - clockwise about the origin, we use the rotation matrix. The rotation matrix for a \(90^{\circ}\) counter - clockwise rotation is \(

$$\begin{bmatrix}0&- 1\\1&0\end{bmatrix}$$

\).
If we have a point \(

$$\begin{bmatrix}x\\y\end{bmatrix}$$

\), after rotation \(

$$\begin{bmatrix}0&-1\\1&0\end{bmatrix}$$

$$\begin{bmatrix}x\\y\end{bmatrix}$$

=

$$\begin{bmatrix}-y\\x\end{bmatrix}$$

\)

Step2: Interpret the result

The resulting point from the matrix multiplication \(

$$\begin{bmatrix}-y\\x\end{bmatrix}$$

\) corresponds to the coordinate \((-y,x)\)

Answer:

\((-y,x)\)