Calculator Inputs
Example Data Table
| Case | Input Type | Input Values | Vector | Magnitude | Unit Vector |
|---|---|---|---|---|---|
| Example 1 | 2D components | (3, 4) | (3, 4) | 5 | (0.6000, 0.8000) |
| Example 2 | 3D components | (2, -1, 2) | (2, -1, 2) | 3 | (0.6667, -0.3333, 0.6667) |
| Example 3 | 2D endpoints | A(1,2), B(6,8) | (5, 6) | 7.8102 | (0.6402, 0.7682) |
Formula Used
For a vector v = [x, y] or v = [x, y, z], first compute magnitude:
|v| = sqrt(x^2 + y^2) for 2D, or |v| = sqrt(x^2 + y^2 + z^2) for 3D.
The unit vector is found by dividing each component by magnitude:
u = v / |v|
When you enter endpoints, the direction vector becomes B - A. Direction cosines are the same as the unit vector components, and axis angles use theta = arccos(u_i).
How to Use This Calculator
- Select whether your input is 2D components, 3D components, 2D endpoints, or 3D endpoints.
- Enter the vector values or the coordinates for points A and B.
- Choose the number of decimal places for output precision.
- Press Find Unit Vector to compute the normalized direction.
- Review the result summary and the component level table above the form.
- Use the CSV or PDF buttons to export the computed results.
Why This Unit Vector Finder Helps
This calculator supports classroom work, coordinate geometry, physics, graphics, and engineering preparation. It accepts either direct components or endpoints, then returns the normalized vector, direction cosines, and axis angles in one place. The responsive layout also makes it easy to review the result table on large and small screens.
FAQs
1. What is a unit vector?
A unit vector has magnitude one. It keeps direction unchanged while removing the original vector length, making comparisons and directional analysis easier.
2. Can this calculator handle 2D and 3D vectors?
Yes. You can switch between 2D and 3D modes, either by entering components directly or by entering start and end points.
3. Why can’t a zero vector have a unit vector?
A zero vector has magnitude zero. Since normalization divides by magnitude, division by zero makes the unit vector undefined.
4. What are direction cosines?
Direction cosines are the cosine values of the angles between the vector and the coordinate axes. They match the unit vector components.
5. Are endpoint inputs converted before normalization?
Yes. The calculator first subtracts point A from point B to create the direction vector, then normalizes that vector.
6. What does the angle output represent?
Each angle shows how the vector is oriented relative to the x, y, or z axis. The value is reported in degrees.
7. When should I export CSV or PDF?
Use CSV for spreadsheet review or batch notes. Use PDF when you want a cleaner shareable report for homework, design checks, or documentation.