Dot Product Calculator – Angle Between Vectors

The dot product multiplies matching components and sums them: for u = (2,3) and v = (4,1), u·v = 2×4 + 3×1 = 11. Its geometric meaning comes from u·v = |u||v|cos θ: with |u| = √13 ≈ 3.606 and |v| = √17 ≈ 4.123, cos θ = 11/14.866 ≈ 0.740, so θ ≈ 42.3°. A zero dot product means cos θ = 0 — the vectors are perpendicular; (2,3)·(−3,2) = −6+6 = 0 confirms a right angle. The calculator lists each component product, the sum, both magnitudes, and the angle, in 2D through 6D. Work applications follow directly: the projection of u onto v equals (u·v/|v|²)v, and physics computes work as F·d — a force of (3,4) moving through (2,0) does 6 units of work.

1. 1

   Enter both vectors

   Matching dimensions, 2D to 6D

2. 2

   Multiply components

   Each uᵢvᵢ product is listed

3. 3

   Sum the products

   u·v = 11 for the example

4. 4

   Extract the angle

   θ = arccos(u·v/(|u||v|)) ≈ 42.3°, with a perpendicularity flag

• 1

  u·v = 0 ⇔ perpendicular (for nonzero vectors)

• 2

  Order does not matter: u·v = v·u

• 3

  The dot product is a scalar, never a vector

• Multiplying mismatched components across different dimensions

• Confusing the dot (scalar) product with the cross (vector) product

• Forgetting the magnitudes when computing the angle

• Reading u·v = 11 as an angle — cos θ = 11/(√13·√17) first