Inputs
Results
–
Angle (acute)
–
Complement (obtuse)
| Quantity | Value |
|---|---|
| Vector v | – |
| Line direction d | – |
| ‖v‖ | – |
| ‖d‖ | – |
| v · d | – |
| |cos θ| | – |
| θ (acute) | – |
| 180° − θ (obtuse) | – |
Formula Used
The angle between a vector v and a line equals the angle between v and the line’s direction vector d.
cos θ = | v · d | / ( ‖v‖ ‖d‖ )
θ is reported as the principal acute angle. The obtuse complement is optional.
cos θ = | v · d | / ( ‖v‖ ‖d‖ )
θ is reported as the principal acute angle. The obtuse complement is optional.
How to Use
- Enter the components of vector v.
- Choose to define the line by a direction vector or two points.
- Select desired angle units.
- Click Compute to calculate the acute angle.
- Enable the checkbox to also show its obtuse complement.
- Use Download CSV or Download PDF to export results.
Example Data Table
Change rows and click Compute to test quickly.
| # | vx | vy | vz | Mode | dx/x₁ | dy/y₁ | dz/z₁ | dx or x₂ | dy or y₂ | dz or z₂ | Angle (deg) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 3 | -2 | 5 | direction | 2 | 1 | -1 | – | |||
| 2 | 1 | 2 | 2 | twopoints | 1 | 0 | 1 | 4 | 3 | 2 | – |
| 3 | -4 | 0 | 7 | direction | 0 | 1 | 1 | – |
FAQs
The principal acute angle between the vector and the line direction. You can also show its obtuse complement, which equals 180° minus the acute angle.
Yes. The direction used is d = P₂ − P₁. Ensure the two points are not identical, otherwise the direction would be the zero vector.
The angle is undefined because ‖v‖ or ‖d‖ equals zero. The calculator will warn you to use nonzero vectors only.
A line has two opposite directions. Using |v·d| removes sign ambiguity and yields the acute angle consistently.
Yes. Select Radians in the Units dropdown. Results and exports will follow the chosen unit.
Use Download CSV for spreadsheets. Use Download PDF for reports or printing. The PDF includes inputs, steps, and the angle result.