3.2-05 Two-dimensional rotations

Two-dimensional rotations — linear algebra practice

Derive rotation matrices from the two standard basis images under a fixed angle. Calculate clockwise and counterclockwise turns, use exact trigonometric values, recover a matrix from a known basis image and interpret inverse and transpose relationships. Angle conventions and input-vector linearity are kept distinct.

What you will practise

• Construct fixed-angle rotation matrices
• Calculate exact rotated vectors
• Explain invariants and inverse-angle behavior

This microlearning belongs to Module 3 of the Complete Linear Algebra Course. Single-select and multiselect questions progress from interpretation to calculation, application and error diagnosis. Each problem is self-contained, and the explanations show how to justify a result rather than only identify a choice. In multiselect tasks, assess each statement separately; the number of correct choices varies.

Before starting, review plane angles, sine, cosine and basis images. Work through the reasoning on paper, compare your method with the explanation and revisit the assumptions behind any missed item. These skills prepare you to compose planar transformations with reliable signs.