Question

Given a point (1,2) on a geometric figure, what is the new point when tje figure is rotated clockwise about the origin 180 degrees

Answer 1

To rotate a point about the origin by a certain angle, we can use the rotation matrix:

\[\begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}\]

In this case, we want to rotate the point (1, 2) clockwise about the origin by 180 degrees. The angle is given in radians by π (since 180 degrees = π radians).

Using the rotation matrix, the new coordinates can be found as:

\[\begin{bmatrix} \cos(π) & -\sin(π) \\ \sin(π) & \cos(π) \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 2 \end{bmatrix}\]

Calculating this, we have:

\[\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} -1 \cdot 1 + 0 \cdot 2 \\ 0 \cdot 1 + (-1) \cdot 2 \end{bmatrix} = \begin{bmatrix} -1 \\ -2 \end{bmatrix}\]

Therefore, the new point after the rotation is (-1, -2).