Given a point (1, 2) on a geometric figure, what is the new point when the figure is rotated counterclockwise about the origin 90 degrees
To rotate a point counterclockwise about the origin 90 degrees, we can use the rotation matrix formula:
\[
\begin{bmatrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{bmatrix}
\begin{bmatrix}
x \\
y
\end{bmatrix}
\]
Where \((x, y)\) is the original point and \(\theta\) is the angle of rotation, which in this case is 90 degrees.
Plugging in \((x, y) = (1, 2)\) and \(\theta = 90^\circ\) (or \(\frac{\pi}{2}\) radians), we have:
\[
\begin{bmatrix}
\cos(\frac{\pi}{2}) & -\sin(\frac{\pi}{2}) \\
\sin(\frac{\pi}{2}) & \cos(\frac{\pi}{2})
\end{bmatrix}
\begin{bmatrix}
1 \\
2
\end{bmatrix}
\]
Evaluating the matrix, we get:
\[
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
\begin{bmatrix}
1 \\
2
\end{bmatrix}
= \begin{bmatrix}
(0)(1) + (-1)(2) \\
(1)(1) + (0)(2)
\end{bmatrix}
= \begin{bmatrix}
-2 \\
1
\end{bmatrix}
\]
Therefore, when the figure is rotated counterclockwise about the origin 90 degrees, the new point is (-2, 1).