Magnitude of a Vector — Definition, Formula & Examples

Magnitude of a Vector
Norm of a Vector

The length of a vector.

Coordinate axes with a vector arrow from origin; labels show "magnitude = length" and "argument = angle

See also

Key Formula

\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}

Where:

$\mathbf{v}$ = A vector with components $v_1, v_2, \ldots, v_n$
$\|\mathbf{v}\|$ = The magnitude (length) of the vector
$n$ = The number of dimensions (components) of the vector

Worked Example

Problem: Find the magnitude of the vector v = ⟨3, 4⟩.

Step 1: Write the magnitude formula for a 2D vector.

\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2}

Step 2: Substitute the components

v_1 = 3

and

v_2 = 4

\|\mathbf{v}\| = \sqrt{3^2 + 4^2}

Step 3: Square each component and add.

\|\mathbf{v}\| = \sqrt{9 + 16} = \sqrt{25}

Step 4: Take the square root.

\|\mathbf{v}\| = 5

Answer: The magnitude of ⟨3, 4⟩ is 5.

Another Example

Problem: Find the magnitude of the 3D vector w = ⟨1, 2, 2⟩.

Step 1: Write the magnitude formula for a 3D vector.

\|\mathbf{w}\| = \sqrt{w_1^2 + w_2^2 + w_3^2}

Step 2: Substitute the components

w_1 = 1

w_2 = 2

, and

w_3 = 2

\|\mathbf{w}\| = \sqrt{1^2 + 2^2 + 2^2}

Step 3: Square each component and add.

\|\mathbf{w}\| = \sqrt{1 + 4 + 4} = \sqrt{9}

Step 4: Take the square root.

\|\mathbf{w}\| = 3

Answer: The magnitude of ⟨1, 2, 2⟩ is 3.

Frequently Asked Questions

Can the magnitude of a vector be negative?

No. Magnitude is always zero or positive. The formula involves squaring each component (which removes negatives) and then taking a positive square root. The only vector with magnitude zero is the zero vector ⟨0, 0, …, 0⟩.

What is the difference between magnitude and direction of a vector?

Magnitude tells you how long the vector is — a single number. Direction tells you where the vector points, often given as an angle or as a unit vector. Together, magnitude and direction completely describe a vector.

Magnitude of a vector vs. Argument (direction) of a vector

Magnitude is a scalar that measures how long a vector is. The argument (or direction angle) measures the angle the vector makes, typically with the positive

x

-axis. A vector like ⟨3, 4⟩ has magnitude 5 and an argument of approximately 53.13°. You need both pieces of information to fully reconstruct the vector from its polar form.

Why It Matters

Magnitude appears throughout physics and engineering whenever you need the size of a quantity that has direction — speed from a velocity vector, strength from a force vector, or distance between two points. In unit vector calculations, you divide a vector by its magnitude to get a direction-only vector of length 1. It also underpins the distance formula in coordinate geometry, since the distance between two points equals the magnitude of the vector connecting them.

Common Mistakes

Mistake: Adding the components instead of squaring them first: writing

\|\langle 3,4\rangle\| = \sqrt{3+4}

Correction: You must square each component before adding. The correct calculation is

\sqrt{3^2 + 4^2} = \sqrt{9+16} = 5

Mistake: Forgetting to take the square root at the end, leaving the answer as the sum of squares.

Correction: The sum of squares gives

\|\mathbf{v}\|^2

, not

\|\mathbf{v}\|

. Always finish by taking the square root to get the actual length.

Related Terms

Vector — The object whose magnitude is measured
Magnitude — General concept of size or absolute value
Argument of a Vector — Direction angle that pairs with magnitude
Unit Vector — Vector with magnitude equal to 1
Distance Formula — Uses the same computation as vector magnitude
Dot Product — Magnitude can be expressed as √(v · v)
Norm — Another name for magnitude of a vector