Given a point ( x, y ), what is the new point when the point is rotated counterclockwise about the origin 270 degrees?(1 point
To rotate a point counterclockwise about the origin, 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.
In this case, the angle of rotation is 270 degrees.
Using the rotation matrix, we can find the new coordinates of the point (x, y) after rotating counterclockwise about the origin 270 degrees:
\[\begin{bmatrix} \cos(270^\circ) & -\sin(270^\circ) \\ \sin(270^\circ) & \cos(270^\circ) \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}\]
Evaluating the trigonometric functions for 270 degrees:
\[\cos(270^\circ) = 0\]
\[\sin(270^\circ) = -1\]
Substituting these values into the matrix equation:
\[\begin{bmatrix} 0 & -(-1) \\ -1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}\]
Simplifying:
\[\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}\]
This results in the new point:
\[(0x + 1y, -1x + 0y)\]
Which simplifies to:
\[(y, -x)\]
Therefore, when the point (x, y) is rotated counterclockwise about the origin 270 degrees, the new point is (-y, x).