Laws of vector arithmetic

Laws of vector arithmetic: linear algebra practice

Use the laws of vector arithmetic as reasoning tools. Rather than recalling names alone, you will identify a law in a calculation, compare valid and invalid rearrangements, simplify an expression and complete a component-based justification. All vectors in an expression belong to the same real coordinate space.

What you will practise

• Distinguish addition and scaling laws from invalid analogies
• Use component arguments to justify distributivity
• Simplify vector expressions and identify failed reasoning

How to use this questionset

Prerequisites: Componentwise addition, scalar multiplication and elementary algebra. 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.

Ordinary vector division and multiplication of two vectors are not assumed. Scalar coefficients and vector operands have different roles. These laws justify the rearrangements used in linear combinations and homogeneous vector equations. This microlearning belongs to Module 1, Vectors and Geometry, in the Complete Linear Algebra Course.