Enter Vector Components
Example Data Table
| Vector | Components | Magnitude | α | β | γ |
|---|---|---|---|---|---|
| A | (3, 4, 12) | 13.0000 | 76.6576° | 72.0798° | 22.6199° |
| B | (-5, 2, 1) | 5.4772 | 155.9052° | 68.5833° | 79.4803° |
| C | (7, -7, 7) | 12.1244 | 54.7356° | 125.2644° | 54.7356° |
| D | (1, 1, 1) | 1.7321 | 54.7356° | 54.7356° | 54.7356° |
Formula Used
These formulas work for any nonzero three-dimensional vector. Negative components naturally produce obtuse direction angles when a component points opposite a positive axis.
How to Use This Calculator
- Enter the x, y, and z components of your vector.
- Add an optional vector label and unit label.
- Choose the number of decimal places you want.
- Click Calculate Angles to compute the results.
- Review magnitude, direction cosines, direction angles, azimuth, elevation, and the unit vector.
- Use the CSV and PDF buttons to export your result summary.
- Inspect the Plotly graph to visualize direction in three-dimensional space.
Frequently Asked Questions
1) What are vector direction angles?
They are the angles a vector makes with the positive x, y, and z axes. They describe spatial orientation using α, β, and γ.
2) Why can a direction angle exceed 90 degrees?
A negative component creates a negative direction cosine. The inverse cosine of a negative value gives an obtuse angle, which is correct.
3) Why is the zero vector invalid here?
A zero vector has no direction. Its magnitude is zero, so direction cosines and direction angles cannot be defined safely.
4) Do the direction cosines follow a rule?
Yes. For any nonzero vector, cos²α + cos²β + cos²γ equals 1, aside from tiny rounding differences.
5) Can I enter decimal or negative values?
Yes. Real vectors often use decimal measurements and may point negatively along one or more coordinate directions.
6) What is the difference between direction angles and azimuth?
Direction angles compare the vector with all three axes. Azimuth tracks only the horizontal heading in the xy-plane.
7) Why does the calculator also show a unit vector?
The unit vector keeps the same direction but reduces magnitude to 1. That makes direction easier to compare and reuse.
8) How should I interpret elevation?
Elevation is the angle above or below the xy-plane. Positive values indicate upward tilt, while negative values indicate downward tilt.