Rotate the point (−4, −6) on a geometric figure 90 degrees counterclockwise. What is the new point?
To rotate a point 90 degrees counterclockwise, we can use the following rotation matrix:
\[\begin{bmatrix} \cos(90^\circ) & -\sin(90^\circ) \\ \sin(90^\circ) & \cos(90^\circ) \end{bmatrix}\]
Since the point we want to rotate is (-4, -6), we substitute these values into the matrix:
\[\begin{bmatrix} \cos(90^\circ) & -\sin(90^\circ) \\ \sin(90^\circ) & \cos(90^\circ) \end{bmatrix} \begin{bmatrix} -4 \\ -6 \end{bmatrix}\]
Evaluating the cosines and sines of 90 degrees gives us:
\[\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} -4 \\ -6 \end{bmatrix}\]
Multiplying the matrices gives us:
\[\begin{bmatrix} 0 \\ 4 \end{bmatrix}\]
Therefore, the new point after a 90-degree counterclockwise rotation is (0, 4).