Vector Magnitude Calculator
Calculate the magnitude (length or Euclidean norm) of a 2D, 3D, or n-dimensional vector. Get step-by-step computation showing each squared component, the sum under the radical, and the final square root result, with an interactive vector diagram.
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About Vector Magnitude Calculator
The Vector Magnitude Calculator computes the length (Euclidean norm) of a vector in any number of dimensions. Enter your vector components and instantly get the magnitude, unit vector, direction angles, a detailed step-by-step square root computation, component contribution analysis, and an interactive diagram showing the vector and its projections.
The Magnitude Formula
For a vector \(\vec{v} = \langle v_1, v_2, \ldots, v_n \rangle\), the magnitude (or Euclidean norm) is:
$$|\vec{v}| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$$
This is a direct generalization of the Pythagorean theorem. In 2D, \(|\vec{v}| = \sqrt{x^2 + y^2}\) gives the hypotenuse of the right triangle formed by the components. In 3D, it extends to \(|\vec{v}| = \sqrt{x^2 + y^2 + z^2}\).
Magnitude by Dimension
2D Vector
\(|\vec{v}| = \sqrt{x^2 + y^2}\)
The length of a vector in the xy-plane. Equivalent to the distance from the origin to point (x, y).
3D Vector
\(|\vec{v}| = \sqrt{x^2 + y^2 + z^2}\)
Used in physics, engineering, and 3D graphics for spatial calculations.
Unit Vector
\(\hat{v} = \frac{\vec{v}}{|\vec{v}|}\)
A vector with magnitude 1 pointing in the same direction. Used to represent pure direction.
n-Dimensional
\(|\vec{v}| = \sqrt{\sum_{i=1}^{n} v_i^2}\)
The same formula works in any dimension, used in data science and machine learning.
Key Formulas Reference
| Formula | Expression | Description |
|---|---|---|
| Magnitude | \(|\vec{v}| = \sqrt{\sum v_i^2}\) | Length of the vector (L2 norm) |
| Unit Vector | \(\hat{v} = \vec{v} / |\vec{v}|\) | Normalized direction vector |
| Direction Cosine | \(\cos \alpha_i = v_i / |\vec{v}|\) | Cosine of angle with each axis |
| Distance | \(d = |\vec{B} - \vec{A}|\) | Distance between two points |
Real-World Applications
How to Use the Vector Magnitude Calculator
- Select a dimension — choose 2D, 3D, or Custom for higher dimensions. Or click a quick example to auto-fill a sample vector.
- Enter components — type the vector components separated by commas (e.g.,
3, 4for 2D or1, 2, 3for 3D). Spaces, semicolons, and international number formats are also supported. - Click Calculate — press the "Calculate Magnitude" button to compute all results.
- Review results — see the magnitude, unit vector, direction angles, step-by-step formulas, component contribution breakdown, and the interactive vector diagram with toggleable layers.
Frequently Asked Questions
Reference this content, page, or tool as:
"Vector Magnitude Calculator" at https://MiniWebtool.com// from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: 2026-04-10
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