When a point $(x, y)$ is rotated counterclockwise about the origin by 270 degrees, the new coordinates can be calculated using a rotation matrix.

The general rotation formula for counterclockwise rotation by $\theta$ degrees is:

$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$

For a 270° rotation, we use the values:

$\cos 270^\circ = 0, \quad \sin 270^\circ = -1$

Thus, the rotation matrix for 270° counterclockwise becomes:

$\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$

Multiplying this matrix by the point $(x, y)$:

$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$

This gives:

$x' = y, \quad y' = -x$

Thus, after rotating the point $(x, y)$ by 270 degrees counterclockwise about the origin, the new point will be:

$(x', y') = (y, -x)$

Example:

If the original point is $(3, 4)$, after a 270° counterclockwise rotation, the new point will be:

$(4, -3)$

Would you like more details or a specific example?

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Here are 5 related questions for further learning:

1. What is the result of rotating the point (2, 3) by 90° counterclockwise about the origin?
2. How do rotation transformations behave for negative angles, such as -90° or -270°?
3. What is the formula for rotating a point around another point, not the origin?
4. How do the coordinates change for a 180° counterclockwise rotation?
5. Can this method of rotation be applied to 3D coordinates, and if so, how?

Tip: To visualize rotations, graph the original and rotated points on a coordinate plane to better understand the transformation.