6.2-01 Orthogonal versus orthonormal

Orthogonal versus orthonormal

Orthogonal vectors need not be unit vectors, and normalizing a family does not automatically make its members perpendicular. These questions sharpen those distinctions through examples, parameter choices and independence arguments.

What you will practise

• Test orthogonal and orthonormal families.
• Normalize vectors without changing direction.
• Handle zero vectors correctly.
• Explain why nonzero orthogonal vectors are independent.

Begin with recognition and interpretation, then work through calculations, error diagnosis and more demanding reasoning. Single-select and multi-select questions ask you to evaluate the stated assumptions as well as the result. Worked explanations connect the answer to a method or a counterexample, so you can check why tempting alternatives fail.

Before you start

Review Euclidean norms, dot products and linear independence. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports coordinate recovery and Gram–Schmidt orthogonalization within Module 6 of the Complete Linear Algebra Course.