Order and associativity

Order and associativity: linear algebra practice

Examine the algebraic rules of cross products rather than transferring ordinary multiplication rules without checking them. Questions test order reversal, scalar factors and distributivity before evaluating both groupings of a three-vector expression. A concrete basis-vector counterexample explains why parentheses are essential even though cross products distribute over vector addition.

What you will practise

• Apply order reversal and bilinearity of the cross product
• Distinguish cross-product distributivity from associativity
• Construct and evaluate a concrete counterexample to associativity

How to use this questionset

Prerequisites: Right-handed basis products, coordinate cross products and vector addition. 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.

A cross product is generally neither commutative nor associative. Special inputs can make two expressions equal without proving a universal identity. Keeping order and parentheses explicit is essential when interpreting scalar triple products and later vector identities. This microlearning belongs to Module 1, Vectors and Geometry, in the Complete Linear Algebra Course.