MathematicsLinear Algebra and TransformationsA-Level

Rotation Matrix (2D) Calculator

This matrix performs a counter-clockwise rotation of a point or vector in 2D Cartesian space by an angle θ about the origin.

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Rotation Angle

Formula first

Overview

The matrix operates on a column vector by multiplying it to transform coordinates while preserving the origin's position and the lengths of vectors. It is a fundamental component of linear transformations, illustrating how geometric operations are expressed as algebraic matrix operations.

Symbols

Variables

= Rotation Angle, x = Initial X-coordinate, y = Initial Y-coordinate

Rotation Angle
Variable
Initial X-coordinate
Variable
Initial Y-coordinate
Variable

Apply it well

When To Use

When to use: Apply this when you need to find the new coordinates of a point after it has been rotated by a specific angle around the origin.

Why it matters: It is essential in computer graphics, game engine development, and robotics for calculating the orientation and movement of objects in 2D space.

Avoid these traps

Common Mistakes

  • Swapping the positions of sine and cosine components.
  • Incorrectly placing the negative sign on the sine in the wrong row or column.
  • Mixing up degrees and radians when performing calculations.

One free problem

Practice Problem

Rotate the point (1, 0) counter-clockwise by 90 degrees. What is the new x-coordinate?

Rotation Angle90
Initial X-coordinate1
Initial Y-coordinate0

Solve for:

Hint: Use cos(90) = 0 and sin(90) = 1 in the rotation matrix formula.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Stewart, J. (2015). Calculus: Early Transcendentals.
  2. Strang, G. (2016). Introduction to Linear Algebra.
  3. A-Level Mathematics Specification: Further Pure Mathematics (Matrices and Transformations)