Rotation Matrices
Rotate vectors using sine and cosine.
Key idea
Rotate vectors using sine and cosine.
In depth
Rotating a point by θ anticlockwise multiplies by [cos θ, −sin θ; sin θ, cos θ]. Composition of rotations adds the angles, and the matrix proves the sum formulas for sine and cosine.
Worked example
Rotating (1, 0) by 90°: (0, 1).
Formula
Real-world uses
- Rotating sprites and 3D models in games
- Robot arm kinematics and image editing tools
Common mistake
Using the matrix for clockwise rotation without flipping the signs of sine.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Rotation Matrices?
Rotate vectors using sine and cosine. It is a topic at the expert level in Trigonometry.
What is the formula for Rotation Matrices?
The key formula is [cosθ −sinθ; sinθ cosθ].
Where is Rotation Matrices used in real life?
Real-world examples: Rotating sprites and 3D models in games; Robot arm kinematics and image editing tools.
What should I learn before Rotation Matrices?
Make sure you are comfortable with Angle Between Vectors (Trig), the previous topic in Trigonometry.
What is a common mistake with Rotation Matrices?
Using the matrix for clockwise rotation without flipping the signs of sine.
How can I practise Rotation Matrices?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Rotation Matrices?
Move on to Trigonometry in Physics next.
Related expert topics in Trigonometry
- Half Angle Formulas – Trig of θ/2.
- Power Reducing Formulas – Reduce squares to first powers of cos 2θ.
- Product-to-Sum Formulas – Convert products into sums.
- Sum-to-Product Formulas – Convert sums into products.
- Triple Angle Formulas – sin3θ and cos3θ in terms of θ.
- Trig and Inverse Trig Compositions – Evaluate expressions like sin(arccos x).