Input
Saved Runs
| # | l | m | n | α° | β° | γ° | Time |
|---|
Example Data
Click a row to load inputs.
| l | m | n | Notes |
|---|---|---|---|
| 1 | 0 | 0 | Along +x ⇒ α=0°, β=90°, γ=90° |
| 0 | 1 | 0 | Along +y ⇒ α=90°, β=0°, γ=90° |
| 0 | 0 | 1 | Along +z ⇒ α=90°, β=90°, γ=0° |
| 0.57735 | 0.57735 | 0.57735 | Equal components ⇒ α=β=γ≈54.7356° |
Formula Used
For a vector v = (x, y, z), its direction cosines are l = x/‖v‖, m = y/‖v‖, n = z/‖v‖ where ‖v‖ = √(x²+y²+z²).
- Angles to the coordinate axes (in degrees): α = arccos(l)·180/π, β = arccos(m)·180/π, γ = arccos(n)·180/π.
- True direction cosines satisfy: l² + m² + n² = 1.
- If inputs are not unit-length, enable Normalize to convert components into valid direction cosines.
How to Use
- Enter l, m, and n values.
- Optionally check Normalize if values are raw components.
- Choose desired decimal places.
- Click Compute Angles to get α, β, γ in degrees.
- Click Add to Saved Runs, then export your table as CSV or PDF.
FAQs
They are the cosines of angles between a vector and the three coordinate axes: l = cos α, m = cos β, n = cos γ, always satisfying l²+m²+n²=1 for a unit-length vector.
If your inputs are components, not cosines, their squared sum may differ from 1. Normalizing divides by the magnitude, producing valid direction cosines for accurate angle computation.
Angles use arccos which requires arguments within [-1,1]. The app clamps extreme values after optional normalization to avoid domain errors, but out-of-range inputs usually indicate non-unit vectors.
Yes. Pick the decimals setting (0–10). Results and saved runs reflect the selected precision so your exported CSV/PDF matches your presentation needs.
No. These are angles to coordinate axes, not internal angles of a triangle. They are independent, constrained only through their cosines squaring to one when combined.