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 by 90 degrees, we can use the rotation matrix:
\[
\begin{bmatrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{bmatrix}
\]
where θ is the angle of rotation. In this case, θ = 90 degrees.
\[
\begin{bmatrix}
\cos(90^\circ) & -\sin(90^\circ) \\
\sin(90^\circ) & \cos(90^\circ)
\end{bmatrix}
\]
Evaluating the trigonometric functions:
\[
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
\]
Multiplying this matrix with the original point (1, 2):
\[
\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}
\begin{bmatrix}
1 \\
2
\end{bmatrix}
=
\begin{bmatrix}
-2 \\
1
\end{bmatrix}
\]
Therefore, the new point after rotating the figure counterclockwise about the origin by 90 degrees is (-2, 1).