Rotate the point (−4, −6) on a geometric figure 90 degrees counterclockwise. What is the new point?(1 point)
Responses
(6, −4)
left parenthesis 6 comma negative 4 right parenthesis
(6, 4)
left parenthesis 6 comma 4 right parenthesis
(−6, −4)
left parenthesis negative 6 comma negative 4 right parenthesis
(−6, 4)
Incorrect it is 6, -4 because it moves Counterclockwise 90 degrees, (once) so it goes from two negatives to a positive and then a negative. so its (6,-4)
(-6, 4)
that is wrong
My apologies for the incorrect response. Let's calculate the correct answer.
To rotate the point (-4, -6) 90 degrees counterclockwise, 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 (90 degrees in this case).
Since we want to rotate counterclockwise, we use a negative angle of rotation:
\[
\begin{bmatrix}
\cos(-90°) & -\sin(-90°) \\
\sin(-90°) & \cos(-90°)
\end{bmatrix}
\]
Simplifying the matrix gives us:
\[
\begin{bmatrix}
0 & 1 \\
-1 & 0
\end{bmatrix}
\]
Applying this matrix to the point (-4, -6):
\[
\begin{bmatrix}
0 & 1 \\
-1 & 0
\end{bmatrix}
\begin{bmatrix}
-4 \\
-6
\end{bmatrix}
=
\begin{bmatrix}
1 \\
-4
\end{bmatrix}
\]
Therefore, the new point after rotating (-4, -6) 90 degrees counterclockwise is (1, -4).