Vector Calculator
Dot product, magnitudes, the angle between in degrees — and in 3D the cross product with its perpendicularity checked live: (u × v) · u must read zero.
Vector Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- 2D or 3D only — both vectors must match dimension
- No unit-vector or projection output (read them off the cards by hand)
In short: u = (3, 4) and v = (1, 0): the dot product is 3·1 + 4·0 = 3, the magnitudes are |u| = 5 and |v| = 1, and the angle between is acos(3/(5·1)) ≈ 53.130102° — the same angle as the 3-4-5 triangle, because it IS that triangle. In 3D the chip (1, 2, 2) × (3, 1, 0) gives cross product (−2, 6, −5): perpendicular to BOTH inputs, with |u × v| = |u||v|sinθ as the area scale.
Formula
u · v = u₁v₁ + u₂v₂ + u₃v₃
cosθ = (u · v)/(|u||v|) · u × v ⊥ both
dot measures agreement; cross measures the parallelogram they span.
Worked Example
- Parse both. two or three components each — both vectors must live in the same dimension.
- Dot and magnitudes. pairwise products summed; each magnitude is the square root of its own dot product.
- Angle, then cross. θ = acos((u·v)/(|u||v|)) in degrees. In 3D the cross product lands perpendicular to both — the page verifies (u×v)·u = 0 live.
The perpendicular chip is the fast test: u = (3, 4), v = (−4, 3) give dot = −12 + 12 = 0 — and zero dot product is the DEFINITION of perpendicular, no protractor needed. The angle card reads 90 exactly.
Strengths & Limits Of This Model
Where this engine is strong
- The perpendicularity of the cross is verified live, not claimed
- The angle law is shown with YOUR numbers plugged in
Where it stops
- No 4D and up
- No scalar triple product
Practical Use Cases
Work in physics
force dot displacement — energy only flows along the motion
Geometry
the angle between any two directions
3D graphics
cross products build the normals lighting depends on
Methodology & Editorial Standards
Vectors parsed from comma/space text (2 or 3 components, matching dimensions enforced). Dot = Σ uᵢvᵢ; magnitudes via their own dot products; θ = acos(dot / (|u||v|)) converted to degrees. In 3D the cross product component law is computed and verified against both inputs (·u = 0, ·v = 0) and against |u||v|sinθ.
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Vector Calculator — 9 Expert FAQs
9 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
What does the dot product measure?
Agreement. Pairwise products summed: positive when the vectors point the same general way, zero when they are perpendicular, negative when they oppose. It also factorizes as |u||v|cosθ — which is why dividing it out gives the angle. One number, three verdicts.
How does the angle formula work?
Rearranged dot product: cosθ = (u·v)/(|u||v|), then acos in degrees. On the default: 3/(5·1) = 0.6 and acos(0.6) ≈ 53.130102° — the exact angle of the 3-4-5 triangle, because (3, 4) makes that triangle with the x-axis and v = (1, 0) IS the x-axis.
Why is the cross product 3D only?
Because its job is to produce the vector PERPENDICULAR to both inputs, and in 2D there is exactly one perpendicular direction (± 90°) — nothing to build. In 3D the cross lands on the normal to the plane the two vectors span, which is why graphics pipelines live on it. The page computes it whenever both inputs carry three components.
What does |u × v| mean?
The area of the parallelogram the two vectors span — the 3D sibling of the determinant story. The page cross-checks the law live: |u × v| should equal |u||v|sinθ, and on the (1, 2, 2) × (3, 1, 0) chip that is √65 ≈ 8.0623 against the angle computed independently. Two roads, one number.
How is the perpendicularity of the cross verified?
Directly: the page computes (u × v) · u and (u × v) · v and both must read zero — that is the DEFINITION of perpendicular, checked on your actual inputs. On the chip: (−2, 6, −5)·(1, 2, 2) = −2 + 12 − 10 = 0.
What does the right-hand rule decide?
The SIGN of the cross product — which of the two perpendicular directions it points. Curl your right hand from u toward v and your thumb gives u × v; swapping the inputs flips the result (v × u = −(u × v)). The mathematics picks a convention; your hand remembers it.
Why must both vectors share a dimension?
Because the dot product pairs components: u₁ with v₁, u₂ with v₂, and a lonely third component has no partner to multiply. The page refuses mismatched inputs with the count — a 2D vector against a 3D one is a typo, not a mathematics problem.
What does a negative dot product tell me?
The angle exceeds 90° — the vectors point generally AGAINST each other. cosθ negative means θ in (90°, 270°), and for directions that means obtuse to opposite. Physically: negative work means the force fought the motion.
Can components be decimals or negatives?
Any real numbers — nothing here assumes integers. Negatives are the whole point (directions matter), decimals ride through in double precision, and the angle prints to the site’s six-place honesty. The only refusals are non-numeric tokens and mismatched dimensions.