Linear Algebra
Dot Product
Multiply two vectors to get their scalar product — and the angle between them. Works for 2D, 3D, and any n-dimensional vector, with every step shown.
Enter your vectors
3D vectors3 components · ⟨1, 2, 3⟩
3 components · ⟨4, -5, 6⟩
a · b =12
Acute angle — pointing a similar way
|a|
3.742
|b|
8.775
angle θ
68.562°
The math works in any dimension — 2D, 3D, or n-D.
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The basics
What is the dot product?
The dot product takes two vectors and returns one number. It captures how much the two vectors point in the same direction — which is why it shows up everywhere from physics work and lighting in graphics to similarity scores in machine learning. There are two ways to write it, and they always agree.
a · b = a₁b₁ + a₂b₂ + … + aₙbₙ
The component form
Multiply matching components and add. This is the form you compute with, and it works in any number of dimensions.
1·4 + 2·(−5) + 4·6 = 18
a · b = |a| |b| cos θ
The geometric form
The same number, read geometrically: the lengths of a and b times the cosine of the angle between them. Rearranged, it gives you that angle.
|a| |b| cos θ → 18
Method
How to calculate it by hand
Three steps, no matter how many dimensions the vectors have.
- 1
Pair up the components
Write the vectors so each component of a sits above the matching component of b. They must have the same length.
a = (1, 2, 4) b = (4, −5, 6) - 2
Multiply each matching pair
Multiply the first components together, the second together, and so on down the line.
1·4, 2·(−5), 4·6 - 3
Add the products
Sum those products. The single number you get is the dot product.
4 + (−10) + 24 = 18
Worked examples
See it across dimensions
The same procedure scales from flat 2D vectors to the higher dimensions used in data and machine learning.
Given a = (3, 4) and b = (2, 1)
Multiply components: (3)(2) and (4)(1)
Evaluate: 6 + 4
a · b = 10
Where it shows up
Why the dot product matters
A single multiplication that quietly powers physics engines, 3D rendering, and modern machine learning. Every entry of a matrix product is one of these sums, which is why the matrix calculator repeats the same row-by-column arithmetic across a whole grid.
W = F · d
Physics
Work is the dot product of force and displacement — only the force along the direction of motion counts.
n · l
Computer graphics
Lighting models use the dot product of a surface normal and the light direction to decide how bright a pixel is.
cos θ = a·b / |a| |b|
Machine learning
Cosine similarity — the backbone of search and recommendation — is the dot product of normalized vectors.
Properties
Rules worth remembering
A handful of clean algebraic rules make the dot product easy to work with.
a · b = b · a
Commutative
Order doesn't matter — swapping the two vectors gives the same number.
a · (b + c) = a · b + a · c
Distributive
The dot product distributes over vector addition, just like ordinary multiplication.
a · b = 0
Perpendicular vectors
If the dot product is zero, the vectors are orthogonal — they meet at a right angle.
a · b = |a| |b|
Parallel vectors
When vectors point the same way, the dot product equals the product of their magnitudes.
(k a) · b = k (a · b)
Scalar multiplication
Scaling one vector by a number scales the dot product by the same factor.
a · a = |a|²
Self dot product
A vector dotted with itself equals its length squared — the basis of vector magnitude.
FAQ
Common questions
The dot product (or scalar product) multiplies two vectors to produce a single number. You multiply each pair of corresponding components and add the results: a·b = a₁b₁ + a₂b₂ + … + aₙbₙ.
Line the vectors up component by component, multiply each matching pair, and add the products. For (1, 2, 4) and (4, −5, 6) that is 1·4 + 2·(−5) + 4·6 = 4 − 10 + 24 = 18.
Its sign tells you how the vectors are oriented. Positive means they point in a broadly similar direction (an acute angle), negative means they point apart (obtuse), and zero means they are perpendicular.
A dot product of zero means the vectors are orthogonal — at a right angle to each other. It is the cleanest test for perpendicularity, no trigonometry required.
No. The dot product pairs up components one-to-one, so both vectors must have the same number of components. A 2D vector and a 3D vector have no valid dot product.
Rearrange the geometric form a·b = |a||b|cos θ to cos θ = (a·b) ÷ (|a||b|), then take the inverse cosine. The calculator shows this angle alongside each result.
The dot product returns a single number (a scalar) and measures alignment. The cross product returns a vector perpendicular to both inputs and measures the area they span — and it only exists in three dimensions.
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Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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