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

Grade 12 Math

Finding the magnitude of a vector is very much the same as determining the length of the hypotenuse of a right triangle – we almost use the exact Pythagorean formula! In this section, we will learn how the horizontal component and the vertical component enable us to algebraically calculate the magnitude of a vector.

12th Grade12th

Vector Operations in Magnitude and Direction Form