AllEasyToolsUniversity
Three-dimensional vectors
Three-dimensional vectors: formula, units and explained result. No account required. Processing stays in your browser.
u·v = Σuᵢvᵢ; cosθ = (u·v)/(|u||v|)
Dot product, cross product and angle; a zero vector has no defined angle.
How to use this calculator
- Identify the quantities and convert them to the units shown.
- Apply the formula first, then compare with the substitution and result.
- Check the assumptions and whether the order of magnitude makes sense.
Levels are approximate topic groupings, not a complete curriculum for every country. Numerical results are rounded and may lose precision for extreme or nearly singular inputs.
A practical example
Use these sample inputs to check the calculation before entering your own values.
- u₁
- 1
- u₂
- 0
- u₃
- 0
- v₁
- 0
- v₂
- 1
- v₃
- 0
Result for this example
- Dot product
- 0
- Cross product
- 0, 0, 1
- Angle
- 90 °