3D Vector Calculator
Dot product, cross product, angle, lengths, projection and the scalar triple product for three-dimensional vectors — every component computed in the open, with the perpendicularity check built in.
Example — a = (1, 2, 3), b = (4, 5, 6)a·b = 32 · a×b = (−3, 6, −3) · θ = 12.93°
How we got it — the example, step by step
- Dot product: 1×4 + 2×5 + 3×6 = 4 + 10 + 18 = 32.
- Cross product: (2×6 − 3×5, 3×4 − 1×6, 1×5 − 2×4) = (−3, 6, −3).
- Lengths: |a| = √14 ≈ 3.7417, |b| = √77 ≈ 8.7750. cos θ = 32 ÷ (3.7417 × 8.7750) = 0.97463, so θ = 12.93°.
Check it backwards
The cross product must be perpendicular to both inputs: (−3, 6, −3)·(1, 2, 3) = −3 + 12 − 9 = 0, and (−3, 6, −3)·(4, 5, 6) = −12 + 30 − 18 = 0. Both zero — geometry kept its promise.
Questions people also ask
What does the scalar triple product mean?
a·(b×c) is the signed volume of the parallelepiped the three vectors span. For (1,2,3), (4,5,6), (7,8,10) it is −3: the box has volume 3, and the sign records the vectors' handedness. Zero would mean all three lie in one plane.