How to Calculate Vector Dot Product

What is Vector Dot Product?

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.

Formula

u · v = |u| × |v| × cos(θ)

v: 0 ⟺ vectors are perpendicular (orthogonal) — 0 ⟺ vectors are perpendicular (orthogonal)

Step-by-Step Guide

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

Input

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

Result

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

Input

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

Result

u·v = 0 — perpendicular ✓

Frequently Asked Questions

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.