What does the transformation matrix for a 90-degree rotation counterclockwise look like?

The transformation matrix for a 90-degree rotation counterclockwise in a two-dimensional Cartesian coordinate system is indeed represented as | 0 -1 | | 1 0 |. This matrix facilitates the rotation of any point in the plane around the origin (0, 0).

When applying this matrix to a point (x, y), the new coordinates (x', y') can be calculated as follows:

x' = 0 * x - 1 * y = -y

y' = 1 * x + 0 * y = x

Thus, a point originally located at (x, y) will be transformed to the point (-y, x) after a 90-degree counterclockwise rotation. This transformation effectively moves points in the plane in a clockwise manner around the origin, which is characteristic of such a rotation.

The structure of the matrix reflects how the axes are reoriented during this rotation. It helps in retaining the overall orientation and distance of points while changing their positions as per the rotational transformation required. This pattern continues to hold true for any point in space when the operation is applied, maintaining the properties of geometric figures through the rotation.