Linear Algebra
Inner Product Calculator
The standard Euclidean inner product — identical to the dot product in ℝⁿ, written in the more general notation linear algebra prefers. Switch to a weighted inner product, and read off the induced norm, distance, and angle, with every step shown.
Enter your vectors
3D spaceStandard Euclidean inner product — identical to the dot product.
3 components · ⟨1, 2, 3⟩
3 components · ⟨4, −5, 6⟩
Inner product
⟨u, v⟩ =12
Acute — vectors point a similar way
‖u‖
3.7417
‖v‖
8.775
angle θ
68.5624°
‖u−v‖
8.1854
Norm, angle, and distance are all induced by the standard inner product.
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The basics
What is the inner product?
An inner product takes two vectors and returns one number that measures how much they point in the same direction. On ordinary coordinate vectors it is exactly the dot product — but the name signals a more general idea: any rule that behaves like a dot product (symmetric, linear, positive) counts, including weighted sums and products of functions. That generality is what lets a single operation define length, distance, and angle in a vector space.
⟨u, v⟩ = u₁v₁ + u₂v₂ + … + uₙvₙ
The component form
Multiply matching components and add. This standard form is the one you compute with, and it works in any number of dimensions.
1·4 + 2·(−5) + 3·6 = 12
⟨u, v⟩ = |u| |v| cos θ
The geometric form
The same number, read geometrically: the lengths of u and v times the cosine of the angle between them. Rearranged, it hands you that angle.
|u| |v| cos θ → 12
Method
How to calculate it by hand
Three steps for the standard inner product — and one extra factor if you are weighting.
- 1
Pair up the components
Write the vectors so each component of u sits above the matching component of v. They must have the same length.
u = (1, 2, 3) v = (4, −5, 6) - 2
Multiply each matching pair
Multiply the first components together, the second together, and so on. For a weighted inner product, also multiply each term by its positive weight wᵢ.
1·4, 2·(−5), 3·6 - 3
Add the products
Sum those products. The single number you get is the inner product.
4 + (−10) + 18 = 12
Worked examples
See it across cases
The same procedure scales from flat 2D vectors to the higher dimensions used in data — and bends to a weighted inner product when components should not count equally.
Given u = (3, 4) and v = (2, 1)
Multiply components: (3)(2) and (4)(1)
Evaluate: 6 + 4
Angle between them: 26.5651°
⟨u, v⟩ = 10
Two flat plane vectors.
Properties
The four defining axioms
An operation only earns the name “inner product” if it obeys these rules. Together they guarantee it induces a sensible length and angle.
⟨u, v⟩ = ⟨v, u⟩
Symmetry
Swapping the two vectors leaves the value unchanged.
⟨a u + b w, v⟩ = a⟨u, v⟩ + b⟨w, v⟩
Linearity
It is additive and scales out constants in each slot.
⟨u, u⟩ ≥ 0
Positive-definite
A vector dotted with itself is never negative.
⟨u, u⟩ = 0 ⟺ u = 0
Definiteness
Only the zero vector has zero length.
Worth knowing
Rules worth remembering
A handful of clean consequences of the axioms make the inner product easy to work with.
⟨u, v⟩ = 0
Orthogonal vectors
If the inner product is zero, the vectors are orthogonal — they meet at a right angle with respect to that inner product.
‖u‖ = √⟨u, u⟩
It defines length
The norm of a vector is the square root of its inner product with itself. Length is a consequence of the inner product, not a separate idea.
cos θ = ⟨u, v⟩ / ‖u‖ ‖v‖
It defines angle
Rearranging the geometric form gives the cosine of the angle between any two nonzero vectors — even in dimensions you cannot picture.
⟨u, v⟩ = Σ wᵢ uᵢ vᵢ
Weighting is allowed
Attaching a positive weight to each component is still a valid inner product. It is how you make some directions count more than others. Written with the weights on the diagonal it becomes a row-by-matrix-by-column product, which the matrix calculator will multiply out for you.
|⟨u, v⟩| ≤ ‖u‖ ‖v‖
Cauchy–Schwarz
The inner product is never larger in size than the product of the two lengths — equality only when the vectors are parallel.
⟨u, u⟩ ≥ 0
Positive-definite
A vector’s inner product with itself is never negative, and is zero only for the zero vector. This is what makes the induced length real and meaningful.
Where it shows up
Why the inner product matters
More general than the dot product, the inner product is the engine behind geometry, statistics, and modern machine learning.
cos θ = ⟨u, v⟩ / ‖u‖ ‖v‖
Machine learning
Cosine similarity — the backbone of search and recommendation — is the inner product of normalized vectors.
⟨x, x⟩ = Var
Statistics
Variance, covariance, and correlation are inner products on centered data, with correlation being the cosine of the angle between two variables.
⟨f, g⟩ = ∫ f g
Function spaces
Defining the inner product as an integral turns functions into vectors — the basis of Fourier series and quantum mechanics.
Reference
Inner product vs dot product
The two terms are often used interchangeably, and on ordinary coordinate vectors they compute the same number. The difference is one of scope.
Dot product
The specific operation on coordinate vectors in ℝⁿ: multiply matching components and add. It is one example of an inner product.
Inner product
The general operation defined by the axioms — symmetric, linear, positive-definite. It covers the dot product, weighted sums, and integrals of functions alike.
FAQ
Common questions
An inner product takes two vectors and returns a single number (a scalar) that measures how much they point in the same direction. The standard Euclidean inner product multiplies each pair of matching components and adds the results: ⟨u, v⟩ = u₁v₁ + u₂v₂ + … + uₙvₙ.
For real vectors in ℝⁿ, the standard inner product is exactly the dot product — the same arithmetic and the same answer. 'Inner product' is the more general term: it names the abstract operation, which can be defined with weights or on spaces of functions, while 'dot product' usually means the standard one on coordinate vectors.
Line the vectors up component by component, multiply each matching pair, and add the products. For u = (1, 2, 3) and v = (4, −5, 6) that is 1·4 + 2·(−5) + 3·6 = 4 − 10 + 18 = 12.
A weighted inner product assigns a positive weight wᵢ to each component before summing: ⟨u, v⟩ = Σ wᵢ·uᵢ·vᵢ. Every weight must be strictly positive so the operation still defines a valid length and angle. With every weight equal to 1 it reduces to the standard inner product.
An inner product of zero means the two vectors are orthogonal — perpendicular with respect to that inner product. It is the cleanest test for a right angle, no trigonometry required, and it underpins ideas like orthogonal bases and projections.
The norm (length) of a vector is ‖u‖ = √⟨u, u⟩, and the angle between two vectors comes from cos θ = ⟨u, v⟩ ÷ (‖u‖‖v‖). So a single inner product gives you length, distance, and angle all at once — that is why it is the foundation of geometry in a vector space.
A real inner product must be symmetric (⟨u, v⟩ = ⟨v, u⟩), linear in each argument, and positive-definite (⟨u, u⟩ ≥ 0, with equality only for the zero vector). Any operation satisfying these rules is a valid inner product and induces a norm and a notion of angle.
No. The standard inner product pairs components one-to-one, so both vectors must have the same number of components. A 2D vector and a 3D vector have no valid standard inner product.
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Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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