Given a point (1,2) on a geometric figure, what is the new point when the figure is rotated counterclockwise about the origin 90°?
To rotate a point counterclockwise about the origin, we can use the rotation matrix:
\[ \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \]
Where \(\theta\) is the angle of rotation. In this case, we want to rotate 90° counterclockwise, so \(\theta = 90°\). Plugging in the values, we have:
\[ \begin{pmatrix} \cos 90° & -\sin 90° \\ \sin 90° & \cos 90° \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \]
To find the new point when the figure is rotated, we multiply this matrix by the coordinates of the original point:
\[ \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 0 \cdot 1 + (-1) \cdot 2 \\ 1 \cdot 1 + 0 \cdot 2 \end{pmatrix} = \begin{pmatrix} -2 \\ 1 \end{pmatrix} \]
Therefore, the new point after rotating (1,2) counterclockwise 90° about the origin is (-2,1).