Vector Magnitude
Length of a vector.
Key idea
Length of a vector.
In depth
The magnitude |v| = √(v₁² + v₂² + ...) is the length of the vector, from the Pythagorean theorem. It is always non-negative and zero only for the zero vector.
Worked example
|(3, 4)| = 5. |(1, 2, 2)| = 3.
Formula
Real-world uses
- Finding the length of a displacement, a speed or the distance between two points in 3D
Common mistake
Adding the components without squaring.
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 Vector Magnitude?
Length of a vector. It is a topic at the intermediate level in Linear Algebra.
What is the formula for Vector Magnitude?
The key formula is |v| = √(v₁²+v₂²).
Where is Vector Magnitude used in real life?
Real-world examples: Finding the length of a displacement, a speed or the distance between two points in 3D.
What should I learn before Vector Magnitude?
Make sure you are comfortable with Scalar Multiplication, the previous topic in Linear Algebra.
What is a common mistake with Vector Magnitude?
Adding the components without squaring.
How can I practise Vector Magnitude?
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 Vector Magnitude?
Move on to Unit Vectors 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.
- Unit Vectors – Vectors of length 1, found by normalising.
- Matrices: Definition – A rectangular array of numbers with m rows and n columns.