Quick answer
The magnitude of a vector is |v| = √(v₁² + v₂² + …). For (3, 4) it is 5.
Enter the components separated by commas. The calculator returns the magnitude.
Related tools: Vector Projection Calculator: Projection of vector a onto vector b. Unit Vector Calculator: Unit vector in the direction of a vector. Vector Subtraction Calculator: Subtract one vector from another.
How to use the Vector Magnitude Calculator
- Type components, for example 3, 4.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Components | Text, such as 3, 4 | — |
Formula and method
Formula
|v| = √(v₁² + v₂² + …)
IEEE-754 double precision; shown to at most 6 decimal places.
Square root of the sum of squared components.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Components = 3, 4.
| Result | Value |
|---|---|
| Magnitude | 5 |
Example 2
Inputs: Components = 1 2 2.
| Result | Value |
|---|---|
| Magnitude | 3 |
Example 3
Inputs: Components = 0, 0.
| Result | Value |
|---|---|
| Magnitude | 0 |
The first line is the headline result (magnitude). IEEE-754 double precision; shown to at most 6 decimal places.
Pythagoras again
In 2D this is the hypotenuse: √(9 + 16) = 5. In 3D, (1, 2, 2) has magnitude √9 = 3. It extends to any number of dimensions.
Applications
Speed from velocity components, force resultants and distance from the origin. The zero vector has magnitude 0. Magnitude is always non-negative.
Length in two and three dimensions
The vector (3, 4) has magnitude √(9 + 16) = 5, and the vector (2, 3, 6) has magnitude √(4 + 9 + 36) = 7. Magnitude is never negative and is zero only for the zero vector.
Dividing a vector by its magnitude gives a unit vector with the same direction and length 1, which is the usual first step in finding directions.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Components: left empty shows "Enter components." beside that field and no result.
Choosing between the Vector Magnitude Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Vector Projection Calculator](/math/vector-projection-calculator/) | vector A and vector B | Projection |
| [Unit Vector Calculator](/math/unit-vector-calculator/) | vector | Unit vector |
| [Vector Subtraction Calculator](/math/vector-subtraction-calculator/) | vector A and vector B | a − b |
| [Vector Calculator](/math/vector/) | vector a and vector b | Sum (a + b) |
| [Slope Calculator](/math/slope-calculator/) | x₁, y₁ and x₂ | Slope |
Background: algebra
Many algebra calculators rest on the same ideas: write the problem in a standard form, identify the coefficients, apply the formula and check the answer by substituting it back. For a quadratic ax² + bx + c = 0 the roots are (−b ± √(b² − 4ac)) ÷ 2a, and the discriminant b² − 4ac decides the kind of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots.
If a = 0 the equation is linear, not quadratic. A sign slip on the −b term shifts both roots by the same amount, so substitute your roots back to check.
Assumptions and limitations
- The zero vector has magnitude 0
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for components to confirm this.
Frequently asked questions
What is the magnitude of (1, 1)?
The vector (1, 1) has magnitude √(1 + 1) = √2 ≈ 1.414.
How is it related to distance?
It's the distance from the origin.
What is a unit vector?
A vector with magnitude 1.
Does the number of components change the method?
No. Square every component, add them, and take the square root, whether there are two, three or more.
Is magnitude the same as length?
Yes. It is the Euclidean length of the vector from the origin to its endpoint.
What does the Vector Magnitude Calculator calculate?
Length (Euclidean norm) of a vector. It returns Magnitude.
What formula does the Vector Magnitude Calculator use?
|v| = √(v₁² + v₂² + …). Square root of the sum of squared components.
What do I need to enter in the vector Magnitude Calculator?
You enter Components.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. The zero vector has magnitude 0.
What is an example of a vector Magnitude result?
With components 3, 4, the magnitude is 5.
Related calculators
- Vector Projection Calculator — Projection of vector a onto vector b.
- Unit Vector Calculator — Unit vector in the direction of a vector.
- Vector Subtraction Calculator — Subtract one vector from another.
- Vector Calculator — Sum, difference, dot product, cross product and angle of two vectors.
- Slope Calculator — Slope, angle, distance and line equation through two points.
- LCM Calculator — Least common multiple of two or more numbers.
Sources
- OpenStax, Precalculus 2e (Rice University): vectors, polar coordinates and trigonometry
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.