What is vector magnitude?

The magnitude (also called the norm or length) of a vector tells you how long the vector is — like the length of an arrow drawn from the origin to a point. In everyday language: if a vector represents a displacement, a velocity, a force, or a direction, the magnitude is the size of that quantity without caring about its direction.

For a 2D vector v = (x, y), the magnitude is:
|v| = √(x² + y²).
This comes straight from the Pythagorean theorem: if you draw x and y as the legs of a right triangle, the magnitude is the hypotenuse.

For a 3D vector v = (x, y, z), you extend the idea into three dimensions:
|v| = √(x² + y² + z²).
And for an N-dimensional vector with components (a₁, a₂, …, aₙ):
|v| = √(a₁² + a₂² + … + aₙ²).

Squares make everything positive, so positive and negative components don’t cancel each other out. Then the square root brings the result back to the same “units” as your original components. For example, if x and y are measured in meters, the magnitude is also in meters.

• |v| is the magnitude (physics-friendly).
• ‖v‖ is the norm (math/linear-algebra-friendly).
• For Euclidean magnitude specifically, it’s often called the L2 norm.

• Physics: speed from velocity, distance from displacement, force size from components.
• Geometry: distance between two points (difference vector magnitude).
• Machine learning: vector length in embeddings, normalization, cosine similarity.
• Engineering: resultant of multiple forces and load components.

How this calculator computes |v| (step-by-step)

This tool follows the exact procedure you’d do by hand — it just does it faster, and formats the steps in a clean way you can screenshot.

The calculator first collects your components. In 2D mode you provide x and y. In 3D mode you provide x, y, z. In N-D mode you can paste a list like 1, -2, 3.5, 0. The tool accepts commas or spaces.

Each component is squared: x², y², z², … This prevents negatives from cancelling positives and ensures the total is non-negative.

The squared values are summed to produce a single number (sometimes called the “squared magnitude”). For v = (3, 4), the sum is 3² + 4² = 9 + 16 = 25.

The square root of the sum gives the magnitude: √25 = 5. If the sum is not a perfect square, you’ll see an exact form (like √29) plus a decimal approximation, depending on your chosen precision.

• Unit vector: v/|v| — gives a vector with the same direction but length 1.
• 2D direction angle: θ = atan2(y, x) in degrees (where defined).
• 3D direction cosines: (x/|v|, y/|v|, z/|v|) — often used in mechanics.

Important edge case: if your vector is the zero vector (all components 0), then |v| = 0, and the unit vector is undefined (division by zero). The calculator will tell you that clearly.

Vector magnitude examples (2D, 3D, and N-D)

Let v = (3, 4). Then:
|v| = √(3² + 4²) = √(9 + 16) = √25 = 5.
The unit vector is v/|v| = (3/5, 4/5) = (0.6, 0.8). This shows the same direction but a length of 1.

Let v = (−6, 8). Squares remove the negative:
|v| = √((−6)² + 8²) = √(36 + 64) = √100 = 10.
Notice how the negative sign doesn’t make the magnitude negative; magnitudes are always ≥ 0.

Let v = (2, −1, 2). Then:
|v| = √(2² + (−1)² + 2²) = √(4 + 1 + 4) = √9 = 3.
Unit vector = (2/3, −1/3, 2/3).

For v = (1, 2, 3, 4):
|v| = √(1² + 2² + 3² + 4²) = √(1 + 4 + 9 + 16) = √30 ≈ 5.4772.
N-dimensional vectors show up in data science and machine learning (feature vectors, embeddings, etc.).

The distance between two points A(x₁, y₁) and B(x₂, y₂) can be computed by taking the difference vector: v = (x₂ − x₁, y₂ − y₁) and then finding |v|.
Example: A(1, 2), B(6, 14). Then v = (5, 12) and |v| = 13. That’s why (5, 12, 13) is a famous triple.

Frequently Asked Questions

• Is vector magnitude the same as absolute value?

  They’re related. Absolute value is magnitude for a 1D number on the number line. Vector magnitude is the “absolute value” of a vector — it tells you how big it is, regardless of direction.

• Can the magnitude of a vector be negative?

  No. Because it’s based on squares and a square root, |v| is always ≥ 0. The only time it equals 0 is for the zero vector (all components 0).

• What does a unit vector mean?

  A unit vector has magnitude 1 and points in the same direction as your original vector. It’s used to separate “direction” from “size.” For example, force = (magnitude) × (direction unit vector).

• What if my teacher wants exact answers?

  Keep the exact radical form when possible (like √30). This calculator shows the squared-sum step so you can present √(sum of squares) as the exact magnitude, then optionally provide a rounded decimal.

• Does this work for complex numbers or other norms?

  This page calculates the standard Euclidean magnitude (L2 norm) for real-number components. Other norms (L1/Manhattan, L∞/max norm) are different formulas — great ideas for additional calculators.

• Why use atan2 for the 2D direction angle?

  atan2(y, x) correctly identifies the quadrant of the vector (so (−1, 1) gives 135° instead of −45°). The calculator uses atan2 to avoid common mistakes.