Vector Input Form
Use components directly or build a vector from two points.
Example Data Table
| Example | Input Type | Vector or Points | Squared Length | Length |
|---|---|---|---|---|
| 2D Navigation | Components | <3, 4> | 25 | 5 |
| 3D Geometry | Components | <2, -1, 2> | 9 | 3 |
| Distance Between Points | Two Points | A(1, 2, 3), B(4, 6, 3) | 25 | 5 |
| 4D Analysis | Components | <1, 2, 2, 1> | 10 | 3.1623 |
Formula Used
|v| = √(x₁² + x₂² + x₃² + ... + xₙ²)
v = B - A = (b₁ - a₁, b₂ - a₂, ... , bₙ - aₙ)
|v|² = x₁² + x₂² + x₃² + ... + xₙ²
||v||₁ = |x₁| + |x₂| + ... + |xₙ|
||v||∞ = max(|x₁|, |x₂|, ... , |xₙ|)
û = v / |v|, valid only when |v| ≠ 0
How to Use This Calculator
- Choose whether you want to enter vector components or two coordinate points.
- Select the number of dimensions, from 2D up to 8D.
- Enter each value in the visible fields. Blank values are treated as zero.
- Set decimal precision and add an optional unit if needed.
- Press Calculate Vector Length to show the result above the form.
- Review the Euclidean length, squared length, Manhattan norm, max norm, unit vector, and axis angles.
- Use the CSV or PDF buttons to export the computed report.
- Inspect the Plotly graph to compare component sizes visually.
Frequently Asked Questions
1. What is vector length?
Vector length, or magnitude, measures how large a vector is. It ignores direction and returns a nonnegative value based on the vector’s components.
2. Can I calculate distance between two points here?
Yes. Choose the two-point mode, enter Point A and Point B, and the calculator first builds the displacement vector, then computes its length.
3. Why is the squared length useful?
Squared length avoids the square root and appears often in algebra, optimization, physics, and machine learning when comparing relative vector sizes efficiently.
4. What happens for a zero vector?
The magnitude becomes zero. A unit vector and direction cosines are undefined because dividing by zero is not allowed.
5. What is the difference between Euclidean and Manhattan norms?
Euclidean norm measures straight-line length. Manhattan norm adds absolute component values. They describe size differently and suit different applications.
6. Does this support higher dimensions?
Yes. This version handles vectors from two dimensions through eight dimensions, which is useful for geometry, statistics, and data science work.
7. What do the axis angles mean?
Each axis angle shows how the vector is oriented relative to one coordinate axis. Smaller angles mean stronger alignment with that axis.
8. Can I export my result for reporting?
Yes. After calculation, use the CSV button for spreadsheet-style output or the PDF button for a portable summary report.