Triangle inequality

Triangle inequality: linear algebra practice

Understand the triangle inequality as a statement about the length of a sum. Practice combines bounds, geometric interpretation, counterexamples and the proof using Cauchy–Schwarz. Particular care is given to equality: collinearity alone is not sufficient when two nonzero vectors point in opposite directions.

What you will practise

• Apply the triangle inequality and reverse triangle bound
• Derive the inequality from dot products and Cauchy–Schwarz
• State the equality condition without overlooking orientation or zero

How to use this questionset

Prerequisites: Norms, dot-product expansion and Cauchy–Schwarz. 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.

All norms are Euclidean. The equality condition must account for both orientation and the possibility that one summand is zero. These estimates support geometric reasoning and later error bounds in numerical work. This microlearning belongs to Module 1, Vectors and Geometry, in the Complete Linear Algebra Course.