Magnitude of a Vector
High School
Definition
A magnitude in mathematics refers to the size of some quantity or object. It helps you compare if something is larger or smaller to another object. For the magnitude of a vector, it tells you the length of the vector. Along with telling you a magnitude, vectors also have a direction.
Worked examples
\(\vec{v} = \langle 3, 4 \rangle \quad |\vec{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)
Use the Pythagorean theorem: square each component, add them, then take the square root.
\(\vec{u} = \langle -6, 8 \rangle \quad |\vec{u}| = \sqrt{({-6})^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)
Negative components are squared, so the magnitude is always non-negative.
\(\vec{w} = \langle 1, -2, 2 \rangle \quad |\vec{w}| = \sqrt{1^2 + ({-2})^2 + 2^2} = \sqrt{1 + 4 + 4} = 3\)
In three dimensions, add the square of all three components before taking the square root.
Common mistakes
- \(|\langle 3, 4 \rangle| = 3 + 4 = 7\) → \(|\langle 3, 4 \rangle| = \sqrt{3^2 + 4^2} = 5\) You must square each component, add them, then take the square root — not just add the components.
- \(|\langle -5, 12 \rangle| = \sqrt{-5^2 + 12^2} = \sqrt{-25 + 144}\) → \(|\langle -5, 12 \rangle| = \sqrt{({-5})^2 + 12^2} = \sqrt{25 + 144} = 13\) Square the entire negative number; \(({-5})^2 = 25\), not \(-25\).
- \(|\langle 0, 0 \rangle| = \)undefined\(\) → \(|\langle 0, 0 \rangle| = 0\) The zero vector has magnitude zero; it's the only vector with no direction but a defined magnitude.
Where you'll use it next
You'll use magnitude to normalize vectors (create unit vectors), calculate distance in coordinate geometry, find dot and cross products, and solve physics problems involving velocity, force, and displacement in multiple dimensions.
Found in 2 StudyPug lessons
Magnitude of a Vector
12th Grade12thGrade 12 Math
Discover how to calculate the magnitude of a vector in 2D and 3D using a simple Pythagorean-based formula, with clear worked examples.
Vector Operations in Magnitude and Direction Form
12th Grade12thGrade 12 Math
Master essential vector operations using magnitude and direction. Learn techniques for addition, subtraction, and resolution of vectors. Apply your skills to real-world problems in physics and engineering.
See also
Reviewed by Pat Cheng, M.Ed. — StudyPug Curriculum Lead · Last updated June 6, 2026