The dot product A · B = a₁b₁ + a₂b₂ + a₃b₃ produces a scalar. It equals |A||B|cos(θ) where θ is the angle between vectors. If A·B = 0, vectors are perpendicular. Dot product measures how much two vectors 'align' and is used in physics (work), ML (cosine similarity), and graphics (Lambertian shading).
How to Use
Enter two vectors:
- 2D: (a₁, a₂) and (b₁, b₂)
- 3D: (a₁, a₂, a₃) and (b₁, b₂, b₃)
- Result: Scalar value computed
Formula
- A · B = Σ(aᵢ × bᵢ)
- A · B = |A||B|cos(θ)
- θ = arccos(A·B / (|A||B|))
Projections
Scalar projection of A onto B: comp_B(A) = A·B/|B|. Vector projection: proj_B(A) = (A·B/|B|²) × B. This decomposes A into parallel and perpendicular components relative to B.
Step-by-Step Instructions
- 1Select 2D or 3D mode.
- 2Enter Vector A components.
- 3Enter Vector B components.
- 4View the dot product scalar.
- 5Check the angle and projections.