u·vVector Dot Product
e.g. 3 4 0
e.g. 2 1 5
The dot product (scalar product) of two vectors produces a scalar — a single number — by summing the products of corresponding components. It encodes both the magnitudes of the vectors and the cosine of the angle between them, making it essential for computing projections, work done by a force, lighting in computer graphics, and similarity in machine learning.
- 1u · v = u₁v₁ + u₂v₂ + u₃v₃ + ... (sum of component products)
- 2u · v = |u| × |v| × cos(θ)
- 3θ = arccos(u·v / (|u|×|v|)) — angle between vectors
- 4u · v = 0 ⟺ vectors are perpendicular (orthogonal)
- 5u · v > 0 ⟺ angle < 90°; u · v < 0 ⟺ angle > 90°
u=(3,4,0), v=(4,3,0)=u·v = 12+12+0 = 24, |u|=5, |v|=5, θ=16.3°
u=(1,0,0), v=(0,1,0) (x and y unit vectors)=u·v = 0 — perpendicular ✓
| Application | Formula / Use |
|---|---|
| Angle between vectors | θ = arccos(u·v / |u||v|) |
| Vector projection | proj_v u = (u·v / v·v) × v |
| Work done by force | W = F · d (force · displacement) |
| Cosine similarity (ML) | sim = u·v / (|u||v|) ∈ [−1, 1] |
| Orthogonality test | u · v = 0 means perpendicular |
| Component along direction | u · v̂ (v̂ is unit vector) |
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