Rotation and Reflection Matrices

Key idea

Matrices that rotate or reflect the plane.

In depth

Rotation by θ: [cos θ −sin θ; sin θ cos θ]. Reflection across the x-axis: [1 0; 0 −1]. Both are orthogonal; rotations have determinant 1 and reflections −1.

Worked example

Rotating (1, 0) by 90° with [0 −1; 1 0] gives (0, 1).

Formula

Real-world uses

• Turning sprites in games and flipping images in photo editing software

Common mistake

Using the wrong sign for the sine terms, which rotates the other way.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related advanced topics in Linear Algebra

• Dot Product – Sum of component products.
• Angle Between Vectors – Cosine from the dot product.
• Orthogonality – Vectors are orthogonal when their dot product is 0.
• Projection – Component of one vector along another.
• Cross Product – A vector perpendicular to two vectors in 3D.
• Lines and Planes in Vector Form – Parametric and normal equations.