Rotation 90 Degrees Counterclockwise Calculator

This calculator performs 90-degree counterclockwise rotation of points, vectors, or shapes in 2D space. It's useful for computer graphics, physics simulations, and geometric transformations.

How to Use This Calculator

To rotate a point or vector 90 degrees counterclockwise:

Enter the original coordinates (x, y)
Click "Calculate"
View the rotated coordinates in the result panel
Use the visualization to understand the transformation

The calculator handles both Cartesian coordinates and vectors. For shapes, you can rotate each vertex individually.

Rotation Formula

For a point (x, y) rotated 90° counterclockwise about the origin, the new coordinates (x', y') are calculated using this transformation matrix:

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

This means:

The new x-coordinate is -y
The new y-coordinate is x

Note: Rotation is performed about the origin (0,0). For rotation about another point, you would first translate the point to the origin, perform the rotation, then translate back.

Worked Examples

Example 1: Rotating a Point

Original point: (3, 4)

Rotated point: (-4, 3)

Explanation: The point (3,4) moves to (-4,3) when rotated 90° counterclockwise.

Example 2: Rotating a Vector

Original vector: (5, -2)

Rotated vector: (2, 5)

Explanation: The vector (5,-2) becomes (2,5) after rotation.

Example 3: Rotating a Shape

Original square vertices: (1,1), (1,-1), (-1,-1), (-1,1)

Rotated vertices: (-1,1), (1,1), (1,-1), (-1,-1)

Explanation: Each vertex is rotated individually, resulting in a 90° counterclockwise rotation of the entire shape.

FAQ

What happens if I rotate a point 90° counterclockwise twice?: Rotating twice will result in a 180° rotation, which is equivalent to multiplying the coordinates by -1. The point (x,y) becomes (-x,-y).
Can I rotate points about a different center than the origin?: Yes, but you need to first translate the point to the origin, perform the rotation, then translate back. This calculator only handles rotation about the origin.
What's the difference between rotating a point and a vector?: From a mathematical perspective, there's no difference - both are represented as coordinates. The interpretation changes based on context: points represent locations, vectors represent directions and magnitudes.
How does this relate to complex numbers?: A 90° counterclockwise rotation of a point (x,y) is equivalent to multiplying the complex number x+yi by i (the imaginary unit).