Dependence in two and three dimensions

Dependence in two and three dimensions: linear algebra practice

Recognize geometric dependence in two and three dimensions. The questions compare pairs, triples and larger collections, including zero vectors and pairs that are individually non-collinear but collectively redundant. Reasoning uses origin-based directions and explicit combinations so the geometry remains visible.

What you will practise

• Diagnose collinear pairs and coplanar triples
• Handle zero vectors and excessive vector counts
• Avoid confusing pairwise non-collinearity with independence of the whole set

How to use this questionset

Prerequisites: Span, scalar multiples and redundant directions. 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.

Coplanarity here means a common plane through the origin for the represented vectors; arbitrary arrowhead planes are a different geometric statement. These examples motivate the homogeneous relation test in the next microlearning. This microlearning belongs to Module 1, Vectors and Geometry, in the Complete Linear Algebra Course.