Calculation Result
Step-by-Step Working
Vector Visualisation
The graph displays the first two components. Three-dimensional results also list axis direction angles.
Calculator Inputs
Calculation History
Select a saved result to restore its mode and inputs.
Formula Used
Vector length measures total size without preserving direction. Different norms emphasise different component behaviours. Select the norm matching your mathematical or engineering task.
How to Use
- Select a calculation mode and vector dimension.
- Enter each component, angle, point, or batch row.
- Choose the norm, precision, rounding, and output format.
- Press Calculate to view results and detailed working.
- Copy, print, share, or export the completed calculation.
Inputs accept decimals, negatives, fractions, and scientific notation. Invalid fields receive clear warnings before any calculation. Export tools preserve the displayed result and steps together.
Example Data
| Example | Vector A | Vector B | Useful mode |
|---|---|---|---|
| 2D vector | 3, 4 | 6, 8 | Magnitude or distance |
| 3D vector | 2, -3, 6 | 1, 4, -2 | Projection or angle |
| 4D vector | 1, 2, 3, 4 | 4, 3, 2, 1 | Comparison or Lp norm |
Common Mistakes
Do not mix incompatible units inside one vector. Avoid normalising a zero vector because division becomes undefined. Match dimensions before comparing, subtracting, or projecting vectors.
Direction angles depend on the chosen coordinate convention. Spherical inclination here is measured downward from positive z. Confirm radians or degrees before entering angular values.
Practical Applications
Vector lengths appear in mechanics, navigation, graphics, optimisation, and data science. Distances measure separation across physical or abstract coordinate spaces. Normalised vectors provide direction while keeping length exactly one.
Frequently Asked Questions
What is a vector length?
Vector length is the magnitude of a vector. It measures size while ignoring direction. Euclidean length uses the square root of squared components.
Can the calculator handle more than three dimensions?
Yes, it supports vectors from two through ten dimensions. Batch rows may also use different dimensions. Comparison modes still require matching component counts.
What is the difference between L1 and L2 norms?
The L1 norm sums absolute component values. The L2 norm uses squared components and a square root. Each norm measures size differently.
What does the maximum norm calculate?
The maximum norm returns the largest absolute component. It is also called the infinity norm. This norm is useful for worst-case component limits.
Can a zero vector be normalised?
No, a zero vector has no defined direction. Its magnitude equals zero. Dividing its components by zero is undefined.
How is distance between two points calculated?
The calculator subtracts matching point coordinates. It forms a displacement vector from those differences. The Euclidean length of that displacement gives distance.
Does the calculator support fractions?
Yes, inputs such as 3/4 are accepted. Mixed numbers should be converted before entry. Output can also display a fraction approximation.
How is the angle between vectors found?
The calculator uses the dot-product relationship. It divides the dot product by both magnitudes. The inverse cosine returns the included angle.
What does numerical tolerance control?
Tolerance controls equality, parallel, and orthogonality checks. Small floating-point differences may otherwise cause false mismatches. Use stricter values for higher precision.