Inner Product: Definition, Axioms + Examples
Learn what an inner product is, the axioms it must satisfy, and how dot products, weighted products, complex vectors, and functions fit the definition.
An inner product is a rule that takes two vectors and returns one scalar while obeying symmetry, linearity, and positive-definiteness. The ordinary dot product is the most familiar example, but inner products can also be weighted, complex-valued, or defined on functions instead of coordinate vectors.
Use the inner product calculator for standard and weighted vector examples with the induced length, distance, and angle shown step by step.
Inner product definition
An inner product gives a vector space its geometry. Once a space has an inner product, you can define length, distance, angle, and orthogonality in a consistent way.
For real vectors, an inner product is usually written with angle brackets:
That scalar is not automatically a length or an angle. It becomes useful because it follows strict rules that make length and angle possible.
The three inner product axioms
For real vector spaces, a rule ⟨u, v⟩ is an inner product only if it satisfies these three conditions.
Symmetry:
⟨u, v⟩ = ⟨v, u⟩
Linearity:
⟨a·u + b·w, v⟩ = a⟨u, v⟩ + b⟨w, v⟩
Positive-definiteness:
⟨v, v⟩ ≥ 0, and ⟨v, v⟩ = 0 only when v = 0Positive-definiteness is the rule that makes length work. A vector paired with itself must never produce a negative value, because its length will be defined from that self-product.
Dot product as an inner product
The standard inner product on real coordinate vectors is the dot product. It multiplies matching components and adds the results.
For example:
So in an introductory linear algebra class, "inner product" often means the same arithmetic as the dot product. The dot product guide covers that standard component formula in more detail.
Weighted inner product
A weighted inner product lets some components count more than others. Each weight must be positive, or the rule may stop defining a valid length.
Here is a simple two-dimensional example.
The first coordinate contributes more because its weight is larger. This is useful when different dimensions should not be treated equally.
Complex inner product
For complex vectors, the inner product uses a conjugate. This keeps ⟨v, v⟩ real and non-negative, which is necessary for length.
The symmetry rule also changes slightly. Instead of plain symmetry, complex inner products use conjugate symmetry.
Some textbooks put the conjugate on the first argument instead of the second. The convention can change, but one side must be conjugated.
Inner product of functions
Inner products are not limited to lists of numbers. Functions can form a vector space too, and their inner product is often defined with an integral.
With this rule, two functions are orthogonal when the integral of their product is zero. This is the idea behind Fourier series: sine and cosine functions can act like perpendicular directions in a function space.
Length, angle, and orthogonality
Once an inner product is chosen, length and angle follow from it.
Length:
‖v‖ = √⟨v, v⟩
Angle:
cos(θ) = ⟨u, v⟩ / (‖u‖‖v‖)
Orthogonal:
⟨u, v⟩ = 0This is why the choice of inner product matters. Change the rule, and you change what counts as length, distance, angle, and perpendicular.
What is not an inner product?
Not every rule that combines two vectors into a number is an inner product. The most common failure is positive-definiteness.
Pair a vector with itself:
Because ⟨v, v⟩ can be negative, this rule cannot define a valid length. It is a bilinear form, but it is not an inner product.
Inner product vs dot product
The dot product is one specific inner product: the standard component-by-component rule on real vectors. Inner product is the general category.
Use "dot product" when the problem is ordinary coordinate-vector arithmetic. Use "inner product" when the problem is about the abstract rule, weighted coordinates, complex vectors, or function spaces.
The takeaway
An inner product is the structure that turns a vector space into geometry. The dot product is the everyday example, but the same axioms also support weighted vectors, complex vectors, and functions. Choose the inner product, and you choose what length and perpendicular mean.
Common questions
An inner product is a rule for multiplying two vectors to get one number in a way that supports geometry. Because it follows the inner product axioms, it lets you define vector length, angle, distance, and orthogonality.
For real coordinate vectors with the standard rule, yes. The dot product is the standard inner product on Rⁿ. Inner product is broader because it also includes weighted rules, complex vector rules, and function-space rules.
For real vectors, an inner product must be symmetric, linear, and positive-definite. In complex vector spaces, symmetry becomes conjugate symmetry so that a vector paired with itself remains real and non-negative.
Positive-definiteness makes length possible. Since length is defined by ‖v‖ = √⟨v, v⟩, the self-product ⟨v, v⟩ must never be negative and must equal zero only for the zero vector.
Two vectors are orthogonal when their inner product is zero. In ordinary real coordinates that means perpendicular. In weighted spaces or function spaces, it means perpendicular according to the chosen inner product.
