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90 Degrees Counterclockwise Around The Origin Calculator

Reviewed by Calculator Editorial Team

This calculator helps you find the new coordinates of a point after rotating it 90 degrees counterclockwise around the origin (0,0). The rotation is a fundamental transformation in coordinate geometry that preserves distances and angles, making it essential for various applications in mathematics, physics, and computer graphics.

How to Use This Calculator

To use the calculator, follow these simple steps:

  1. Enter the original x-coordinate of your point in the first input field.
  2. Enter the original y-coordinate of your point in the second input field.
  3. Click the "Calculate" button to perform the rotation.
  4. The calculator will display the new coordinates after rotation.
  5. Use the "Reset" button to clear all inputs and results.

The calculator provides an immediate result and also shows the step-by-step transformation process, helping you understand how the rotation works.

Formula Explained

Rotating a point 90 degrees counterclockwise around the origin involves a specific mathematical transformation. The new coordinates (x', y') of a point (x, y) after rotation can be calculated using the following formulas:

Rotation Formulas

For a 90° counterclockwise rotation around the origin:

x' = -y

y' = x

These formulas work because a 90° counterclockwise rotation swaps the x and y coordinates and changes the sign of the new x-coordinate. This transformation preserves the distance from the origin and maintains the relative positions of all points.

Worked Examples

Let's look at a couple of examples to see how the rotation works in practice.

Example 1: Simple Point

Consider the point (3, 4). To rotate it 90° counterclockwise around the origin:

  1. Original coordinates: (3, 4)
  2. Apply the rotation formulas:
    • x' = -4 = -4
    • y' = 3 = 3
  3. New coordinates after rotation: (-4, 3)

Example 2: Negative Coordinates

Now let's take the point (-2, 5) and rotate it:

  1. Original coordinates: (-2, 5)
  2. Apply the rotation formulas:
    • x' = -5 = -5
    • y' = -2 = -2
  3. New coordinates after rotation: (-5, -2)

Visualizing the Rotation

The rotation moves points along a circular path centered at the origin. Each point maintains the same distance from the origin but changes its angle by 90° counterclockwise.

Visualization

The following chart shows the original point and its rotated position:

The blue dot represents the original point, and the green dot shows its position after rotation. The dashed line indicates the rotation path.

Frequently Asked Questions

What happens if I rotate a point 90° clockwise instead?
The formulas for clockwise rotation are x' = y and y' = -x. This is the inverse of the counterclockwise rotation.
Can I rotate points around other centers besides the origin?
Yes, but the formulas become more complex. You would need to translate the point to the origin, perform the rotation, and then translate back.
Is this rotation used in real-world applications?
Yes, 90° rotations are commonly used in computer graphics, image processing, and navigation systems to change the orientation of objects or coordinate systems.
What if I enter non-numeric values in the calculator?
The calculator will display an error message. Please ensure you enter valid numbers for both coordinates.