Dot Product Calculator - Find u·v Scalar Product

Use this Dot Product Calculator to compute the scalar product of two vectors. The dot product (also called the scalar product) of two vectors u and v is a single number defined by u · v = Σ uᵢvᵢ. In geometric terms, u · v = |u||v| cos θ, where θ is the angle between the vectors. A zero dot product indicates orthogonality, while a maximal magnitude indicates parallel vectors.

This calculator supports 2D, 3D, and custom n‑dimensional vectors. You can enter vector components or use magnitudes with an angle in degrees or radians. Results include magnitudes, the angle between vectors, and scalar projections comp_v(u) and comp_u(v).

Algebraic: u · v = u₁v₁ + u₂v₂ + … + uₙvₙ

Geometric: u · v = |u||v| cos θ

Scalar projection of u onto v: comp_v(u) = (u · v) / |v|

Scalar projection of v onto u: comp_u(v) = (u · v) / |u|

Let u = ⟨1, −2, 3⟩ and v = ⟨4, 0, −1⟩. Then u · v = 1·4 + (−2)·0 + 3·(−1) = 4 + 0 − 3 = 1. The magnitudes are |u| = √(1² + (−2)² + 3²) = √14 and |v| = √(4² + 0² + (−1)²) = √17.

The angle satisfies cos θ = (u · v)/(|u||v|) = 1/(√14 √17). Therefore θ ≈ arccos(1/(√238)) ≈ 86.28°. Scalar projections: comp_v(u) = (u · v)/|v| = 1/√17 and comp_u(v) = (u · v)/|u| = 1/√14.

The dot product is fundamental in physics (work = F · d), projections, measuring similarity, and determining orthogonality. For complete projection breakdown (vector and rejection), use our Vector Projection Calculator. For perpendicular vectors and areas, see the Cross Product Calculator.

When vectors represent directions and magnitudes, the dot product answers “how aligned are these two directions?” A positive value implies acute alignment, zero means orthogonal, and a negative value indicates pointing mostly opposite. In machine learning, cosine similarity normalizes this idea to compare text or embedding vectors independent of magnitude. In physics, work performed by a force along a displacement is computed with exactly this operation: W = F · d.

Common mistakes include mixing units (meters with centimeters), forgetting to convert degrees to radians in trigonometric forms, and comparing vectors in different coordinate frames. If your inputs come from sensors or simulations, ensure they’re expressed in the same basis before computing u · v. The Dot Product Calculator helps by offering both component and magnitude–angle modes, so you can cross‑check results quickly when unit conversions are involved.

• Normalize vectors first to compare direction only (cosine similarity).
• Verify axes and frames match when combining measurements.
• Use higher precision for near‑orthogonal vectors to avoid rounding noise.
• Prefer component mode when values are small or span several orders of magnitude.

Interpreting results matters as much as computing them. A dot product near zero means the directions are nearly perpendicular, so one vector contributes little to the other. Large positive values indicate strong alignment; large negative values mean the vectors point mostly opposite. In physics, this maps directly to work: if F = ⟨12, −5, 0⟩ newtons and d = ⟨0.4, −0.1, 0⟩ meters, then W = F · d = 12·0.4 + (−5)·(−0.1) = 4.8 + 0.5 = 5.3 J. If the displacement were ⟨0.4, 0.1, 0⟩ instead, the work would drop to 12·0.4 + (−5)·0.1 = 4.8 − 0.5 = 4.3 J, illustrating how orientation alone changes the outcome. Use the Dot Product Calculator to explore these “alignment” effects interactively and to sanity‑check projections in real data.