Unit vectors

Unit vectors: linear algebra practice

Practise constructing a unit vector along a given nonzero direction. The questions require exact normalization, check both length and orientation, and distinguish general unit vectors from standard coordinate directions. Later items examine opposite directions and the zero-vector exception.

What you will practise

• Normalize a nonzero vector while preserving direction
• Distinguish unit vectors from standard basis vectors
• Explain why zero cannot be normalized and how negative scaling affects direction

How to use this questionset

Prerequisites: Euclidean norm, scalar multiplication and exact fractions. 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 same-direction unit vector and an opposite-direction unit vector are different answers even though both have length one. Unit directions are used in scalar projection, signed distances and geometric orientation. This microlearning belongs to Module 1, Vectors and Geometry, in the Complete Linear Algebra Course.