Matematika

Skalární součin vektorů

u·vVector Dot Product

Vector u (space separated)

e.g. 3 4 0

Vector v

e.g. 2 1 5

What is Vector Dot Product?

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Bodový součin (skalární součin) dvou vektorů vytvoří skalár — jediné číslo — sečtením součinů odpovídajících složek. Kóduje jak velikosti vektorů, tak kosinus úhlu mezi nimi, takže je nezbytný pro výpočetní projekce, práci vykonávanou silou, osvětlení v počítačové grafice a podobnost ve strojovém učení.

Calkulon makes complex calculations simple — built for students and everyday problem-solvers.

Vzorec

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f(x)u · v = |u| × |v| × cos(θ)

Variable Legend

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Symbol	Jméno	Jednotka	Popis
v	0 ⟺ vectors are perpendicular (orthogonal)	—	0 ⟺ vectors are perpendicular (orthogonal)

How to Vector Dot Product

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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°

Worked Examples

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Example 1

Given:u=(3,4,0), v=(4,3,0)

Výsledek:u·v = 12+12+0 = 24, |u|=5, |v|=5, θ=16.3°

Example 2

Given:u=(1,0,0), v=(0,1,0) (x and y unit vectors)

Výsledek:u·v = 0 — perpendicular ✓

Real-World Applications

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Secondary and university physics/chemistry coursework

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Laboratory calculations and experimental data analysis

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Engineering design and materials science applications

Dot Product Applications

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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)

Frequently Asked Questions

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What is Vector Dot?

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

How accurate is the Vector Dot calculator?

The calculator uses the standard published formula for vector dot. Results are accurate to the precision of the inputs you provide. For financial, medical, or legal decisions, always verify with a qualified professional.

What units does the Vector Dot calculator use?

This calculator works with inches. You can enter values in the units shown — the calculator handles all conversions internally.

What formula does the Vector Dot calculator use?

The core formula is: u · v = u₁v₁ + u₂v₂ + u₃v₃ + ... (sum of component products). Each step in the calculation is shown so you can verify the result manually.

References

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Reviewed October 2026

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