Given a point (-6, 4) in a line segment, what is the new point when the line segment is
rotated counterclockwise about the origin 180 degrees? (1 point)
To rotate a point counterclockwise about the origin, we can use the rotation matrix:
\[\begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}\]
Where \(\theta\) is the angle of rotation. In this case, we want to rotate 180 degrees, so \(\theta = 180^\circ\).
Using this matrix and plugging in the point (-6, 4), we get:
\[\begin{bmatrix} \cos(180^\circ) & -\sin(180^\circ) \\ \sin(180^\circ) & \cos(180^\circ) \end{bmatrix} \begin{bmatrix} -6 \\ 4 \end{bmatrix}\]
Calculating the matrix multiplication:
\[\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} -6 \\ 4 \end{bmatrix} = \begin{bmatrix} (-1)(-6) + (0)(4) \\ (0)(-6) + (-1)(4) \end{bmatrix} = \begin{bmatrix} 6 \\ -4 \end{bmatrix}\]
So the new point after rotating counterclockwise about the origin 180 degrees is (6, -4).