Unit Vectors
Vectors of length 1, found by normalising.
Key idea
Vectors of length 1, found by normalising.
In depth
A unit vector has length 1 and records pure direction. Normalise any non-zero vector by dividing by its length: û = v/|v|. The standard unit vectors i, j, k point along the axes.
Worked example
v = (3, 4): û = (3/5, 4/5).
Formula
Real-world uses
- Giving directions for robots and navigation
- Normalising data before comparing features in machine learning
Common mistake
Dividing by the components instead of the magnitude.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Unit Vectors?
Vectors of length 1, found by normalising. It is a topic at the intermediate level in Linear Algebra.
What is the formula for Unit Vectors?
The key formula is û = v/|v|.
Where is Unit Vectors used in real life?
Real-world examples: Giving directions for robots and navigation; Normalising data before comparing features in machine learning.
What should I learn before Unit Vectors?
Make sure you are comfortable with Vector Magnitude, the previous topic in Linear Algebra.
What is a common mistake with Unit Vectors?
Dividing by the components instead of the magnitude.
How can I practise Unit Vectors?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Unit Vectors?
Move on to Dot Product next.
Related intermediate topics in Linear Algebra
- Scalars and Vectors – A scalar is a number; a vector has magnitude and direction.
- Vector Notation and Components – Write vectors as ordered lists of coordinates.
- Vector Addition – Add component-wise; geometrically tip-to-tail.
- Scalar Multiplication – Scale a vector’s length.
- Vector Magnitude – Length of a vector.
- Matrices: Definition – A rectangular array of numbers with m rows and n columns.