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?
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}$$\begin{bmatrix} x \\ y \\\end{bmatrix}$
where $\theta$ is the angle of rotation.
In this case, we want to rotate the point $(-6, 4)$ counterclockwise about the origin 180 degrees, so $\theta = 180^\circ$.
Plugging in the values, we get:
$\begin{bmatrix} \cos 180^\circ & -\sin 180^\circ \\\sin 180^\circ & \cos 180^\circ \\\end{bmatrix}$$\begin{bmatrix} -6 \\ 4 \\\end{bmatrix}$
Simplifying,
$\begin{bmatrix} -1 & 0 \\ 0 & -1 \\ \end{bmatrix}$$\begin{bmatrix} -6 \\ 4 \\\end{bmatrix}$
We multiply the matrices:
$\begin{bmatrix} -1(-6) + 0(4) \\ 0(-6) + (-1)(4) \\\end{bmatrix}$
Simplifying,
$\begin{bmatrix} 6 \\ -4 \\\end{bmatrix}$
Therefore, the new point after rotating the line segment counterclockwise about the origin 180 degrees is $(6, -4)$.