3D Vector Calculator – Dot & Cross Product
3D Vector Calculator & Cross Product in Oxyz [u, v]
Enter the coordinates of two vectors in Oxyz space. The solution finds the magnitudes, the dot product, the angle between the vectors and the cross product via 2×2 determinants.
The calculator above computes the length of vectors, the dot product, the angle between two vectors, and the cross product in three-dimensional space.
Formulas used
u · v = x₁x₂ + y₁y₂ + z₁z₂
cos φ = (u · v) / (|u|·|v|)
u × v = (y₁z₂ − z₁y₂; z₁x₂ − x₁z₂; x₁y₂ − y₁x₂)
- The cross product is perpendicular to both vectors.
- |u × v| equals the area of the parallelogram built on u and v.
Worked example
u = (1; 0; 0), v = (0; 1; 0):
u · v = 0, u × v = (0; 0; 1)
Enter the same values in the calculator above to see every step of the solution.
How to use the calculator
- Enter the coordinates of vectors u and v.
- Press the calculate button. The result appears instantly.
- Read the step-by-step solution, rendered with KaTeX, to check your own work.