Geometric proofs with dot products

Geometric proofs with dot products: linear algebra practice

Use vector algebra to justify familiar geometry rather than relying on an accurate-looking drawing. Questions examine diagonals of a parallelogram, the rhombus condition, the parallelogram law and the cosine law. The progression moves from useful expansions to a converse argument and a numerical application.

What you will practise

• Use dot-product identities to prove geometric statements
• Relate rhombus diagonals to equal side lengths
• Derive the cosine law and check a converse with the correct hypotheses

How to use this questionset

Prerequisites: Dot products, vector sums and differences, norms and the angle formula. The questions progress from direct interpretation to application and reasoning. Work out an answer before comparing the choices; for multi-select questions, evaluate every statement independently. Worked explanations show the reasoning and address relevant misconceptions, so an incorrect attempt can guide your next review.

Parallelograms described as geometric figures are nondegenerate, with non-collinear adjacent side vectors. These proofs demonstrate how algebra can establish geometry independently of scale or orientation in a diagram. This microlearning belongs to Module 1, Vectors and Geometry, in the Complete Linear Algebra Course.