6.6-03 Pythagoras and the parallelogram law

Pythagoras and the parallelogram law

Pythagoras and the parallelogram law follow from inner-product expansions. The questions combine calculations with converse statements and a concrete norm that fails the parallelogram identity.

What you will practise

• Expand norms and identify cross terms.
• Apply and reverse the real Pythagorean relation.
• Use the parallelogram law to recover missing quantities.
• Distinguish inner-product norms from more general norms.

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 inner-product bilinearity and induced norms. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports energy decompositions and geometric characterization of norms within Module 6 of the Complete Linear Algebra Course.