Calculate the magnitude of a 2D or 3D vector with squared magnitude, unit vector, and direction angles.
Last updated
Dimension
What is vector magnitude?
The magnitude of a vector is its length in Euclidean space, calculated as the square root of the sum of its squared components.
Unit vectors
A unit vector points in the same direction but has a magnitude of exactly 1. It is found by dividing each component by the magnitude.
Result
The magnitude of vector [3, 4] is 5.
Steps
Components: x=3, y=4
Squared: x²=9 + y²=16
Sum of squares: 25
Magnitude: √25 = 5
Unit vector: [0.600000, 0.800000]
Direction angles: α=53.1301°, β=36.8699°
Linear Algebra
A vector magnitude calculator computes the length (norm) of a 2D or 3D vector using the Euclidean distance formula. It also provides the unit vector and direction angles.
The magnitude of vector v = (x, y, z) is |v| = √(x² + y² + z²). For 2D vectors, omit the z component.
The unit vector is v/|v|, pointing in the same direction with magnitude 1. Direction angles give the angle to each coordinate axis.
|v| = √(x² + y² + z²)
Euclidean vector magnitude. This is the specific relationship the calculator applies when building the result.
A worked example helps translate the vector magnitude calculator maths into a realistic scenario so the user can compare the headline result with a concrete set of inputs.
That matters because a result is easier to trust when the page shows how the same logic behaves in a practical case instead of leaving the formula abstract.
Use the vector magnitude calculator output as a planning aid, then compare it with the assumptions, units, and caveats shown elsewhere on the page before acting on the number alone.
That extra interpretation step matters because a calculator can simplify the arithmetic but still cannot replace real-world context such as local rules, contract terms, or individual circumstances.
Frequently asked questions
No — magnitude is always non-negative. It is zero only for the zero vector.
For Euclidean vectors, magnitude and norm (specifically the L2 norm) are the same thing.
The safest manual check is to follow the same formula or rule one step at a time and compare that working with the calculator output. That catches sign errors, bracket mistakes, and input-order mixups without requiring any extra method beyond the underlying maths itself.
Also in Linear Algebra
Related
These related calculators come from the same leaf category, nearby sibling categories, or the same top-level topic.
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Perform vector operations including addition, subtraction, dot product, cross product, and scalar multiplication.
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