Inner Product: Dot Product, Norms & Angles

The Dot Product

The dot product of two vectors

\mathbf{u} = (u_1, \dots, u_n)

and

\mathbf{v} = (v_1, \dots, v_n)

in

\mathbb{R}^n

is

\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + \cdots + u_n v_n = \mathbf{u}^T\mathbf{v}

The result is a scalar, not a vector. The matrix form

\mathbf{u}^T\mathbf{v}

treats

\mathbf{u}

as a

1 \times n

row and

\mathbf{v}

as an

n \times 1

column, making the dot product a

1 \times 1

matrix multiplication.

For

\mathbf{u} = (2, -1, 3, 0)

and

\mathbf{v} = (1, 4, -2, 5)

:

\mathbf{u} \cdot \mathbf{v} = 2(1) + (-1)(4) + 3(-2) + 0(5) = 2 - 4 - 6 + 0 = -8

.

The dot product is the standard inner product on

\mathbb{R}^n

. It is the measuring tool that defines lengths, angles, distances, and perpendicularity throughout finite-dimensional linear algebra.

Paired products collapsed to one number

Matching components were multiplied and the results summed into a single scalar. Everything else on this page — length, distance, angle, orthogonality — is extracted from that one number, which is why the dot product is treated as the primitive and the rest as consequences. Follow the accumulation on the inner product visualizer.

The properties listed next — symmetry, linearity, positivity — are exactly what a general inner product is required to keep.

Properties of the Dot Product

The dot product satisfies three fundamental properties.

Symmetry:

\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}

. The order does not matter.

Linearity:

(c\mathbf{u} + d\mathbf{w}) \cdot \mathbf{v} = c(\mathbf{u} \cdot \mathbf{v}) + d(\mathbf{w} \cdot \mathbf{v})

. The dot product distributes over addition and pulls scalars out. Combined with symmetry, it is linear in both arguments (bilinear).

Positive definiteness:

\mathbf{v} \cdot \mathbf{v} \geq 0

for all

\mathbf{v}

, with equality if and only if

\mathbf{v} = \mathbf{0}

. The quantity

\mathbf{v} \cdot \mathbf{v} = v_1^2 + v_2^2 + \cdots + v_n^2

is a sum of squares, which is zero only when every component is zero.

These three properties — symmetry, linearity, positive definiteness — are not just useful observations. They are the axioms that define an inner product in the abstract setting.

Length

The length (or norm) of a vector

\mathbf{v}

is

\|\mathbf{v}\| = \sqrt{\mathbf{v} \cdot \mathbf{v}} = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}

This is the Euclidean norm — the generalization of the Pythagorean theorem to

n

dimensions. In

\mathbb{R}^2

,

\|(3, 4)\| = \sqrt{9 + 16} = 5

. In

\mathbb{R}^3

,

\|(1, 2, 2)\| = \sqrt{1 + 4 + 4} = 3

.

The norm satisfies

\|\mathbf{v}\| \geq 0

with equality only for

\mathbf{v} = \mathbf{0}

, and

\|c\mathbf{v}\| = |c|\|\mathbf{v}\|

— scaling a vector scales its length by the absolute value of the scalar.

A unit vector has

\|\mathbf{v}\| = 1

. Any nonzero vector can be normalized — replaced by

\hat{\mathbf{v}} = \mathbf{v}/\|\mathbf{v}\|

, which points in the same direction with length

1

. Normalization preserves direction and discards magnitude.

A vector dotted with itself, then rooted

Taking the dot product of a vector with itself gives the sum of its squared components, and the square root of that is its length. Length is therefore not an extra definition bolted on: it is the inner product applied to a single vector. Run it on your own vector on the magnitude visualizer.

Distance in the following section is this same length applied to the difference of two vectors.

Distance

The distance between two vectors

\mathbf{u}

and

\mathbf{v}

is the length of their difference:

Distance Formula

d(\mathbf{u}, \mathbf{v}) = \|\mathbf{u} - \mathbf{v}\|

Distance formula →

Expanded coordinate-wise, this becomes

d(\mathbf{u}, \mathbf{v}) = \sqrt{(u_1 - v_1)^2 + (u_2 - v_2)^2 + \cdots + (u_n - v_n)^2}

the Euclidean distance — the straight-line separation between two points in

\mathbb{R}^n

.

Distance satisfies the properties of a metric:

d(\mathbf{u}, \mathbf{v}) \geq 0

with equality only when

\mathbf{u} = \mathbf{v}

;

d(\mathbf{u}, \mathbf{v}) = d(\mathbf{v}, \mathbf{u})

(symmetry); and

d(\mathbf{u}, \mathbf{w}) \leq d(\mathbf{u}, \mathbf{v}) + d(\mathbf{v}, \mathbf{w})

(triangle inequality). Every inner product induces a distance through this formula, and the resulting distance always satisfies these metric properties.

Angle Between Vectors

For nonzero vectors

\mathbf{u}

and

\mathbf{v}

, the angle

\theta

between them satisfies

\cos\theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}

The Cauchy-Schwarz inequality (next section) guarantees that the right-hand side lies between

-1

and

1

, so the formula always produces a well-defined angle

\theta \in [0, \pi]

.

When

\cos\theta = 1

(

\theta = 0

), the vectors point in the same direction. When

\cos\theta = -1

(

\theta = \pi

), they point in opposite directions. When

\cos\theta = 0

(

\theta = \pi/2

), the vectors are orthogonal.

The orthogonality condition

\mathbf{u} \cdot \mathbf{v} = 0

is the case

\theta = 90°

. The dot product encodes the full metric geometry of

\mathbb{R}^n

: length from

\mathbf{v} \cdot \mathbf{v}

, angle from

\mathbf{u} \cdot \mathbf{v}

, and distance from

(\mathbf{u} - \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v})

.

For

\mathbf{u} = (1, 2, 3)

and

\mathbf{v} = (4, -1, 2)

:

\mathbf{u} \cdot \mathbf{v} = 4 - 2 + 6 = 8

,

\|\mathbf{u}\| = \sqrt{14}

,

\|\mathbf{v}\| = \sqrt{21}

. So

\cos\theta = 8/\sqrt{294} \approx 0.467

, giving

\theta \approx 62.2°

.

The Cauchy-Schwarz Inequality

For all vectors

\mathbf{u}

and

\mathbf{v}

in

\mathbb{R}^n

:

Cauchy-Schwarz Inequality

|\mathbf{u} \cdot \mathbf{v}|\leq \|\mathbf{u}\|\, \|\mathbf{v}\|

Cauchy schwarz inequality →

Equality holds if and only if one vector is a scalar multiple of the other — they are parallel.

The proof considers the expression

\|\mathbf{u} - t\mathbf{v}\|^2 \geq 0

for all

t \in \mathbb{R}

. Expanding:

\|\mathbf{u}\|^2 - 2t(\mathbf{u} \cdot \mathbf{v}) + t^2\|\mathbf{v}\|^2 \geq 0

. This is a quadratic in

t

that is non-negative everywhere, so its discriminant must be non-positive:

4(\mathbf{u} \cdot \mathbf{v})^2 - 4\|\mathbf{u}\|^2\|\mathbf{v}\|^2 \leq 0

. Rearranging gives Cauchy-Schwarz.

The inequality is what makes the angle formula legitimate. It guarantees

-1 \leq \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|} \leq 1

, so

\cos\theta

takes a valid value. Without Cauchy-Schwarz, the angle formula could produce numbers outside

[-1, 1]

, and the geometric interpretation would collapse.

The Triangle Inequality

For all vectors

\mathbf{u}

and

\mathbf{v}

in

\mathbb{R}^n

:

Triangle Inequality

\|\mathbf{u} + \mathbf{v}\|\leq \|\mathbf{u}\|+ \|\mathbf{v}\|

Triangle inequality →

The length of one side of a triangle never exceeds the sum of the other two. Equality holds if and only if

\mathbf{u}

and

\mathbf{v}

point in the same direction (one is a non-negative scalar multiple of the other).

The proof follows from Cauchy-Schwarz. Square both sides:

\|\mathbf{u} + \mathbf{v}\|^2 = \|\mathbf{u}\|^2 + 2\mathbf{u} \cdot \mathbf{v} + \|\mathbf{v}\|^2 \leq \|\mathbf{u}\|^2 + 2\|\mathbf{u}\|\|\mathbf{v}\| + \|\mathbf{v}\|^2 = (\|\mathbf{u}\| + \|\mathbf{v}\|)^2

. The key step uses

\mathbf{u} \cdot \mathbf{v} \leq |\mathbf{u} \cdot \mathbf{v}| \leq \|\mathbf{u}\|\|\mathbf{v}\|

.

The triangle inequality is essential for the distance function

d(\mathbf{u}, \mathbf{v}) = \|\mathbf{u} - \mathbf{v}\|

to satisfy the metric axioms. It ensures that going from

\mathbf{u}

to

\mathbf{w}

directly is never longer than going via

\mathbf{v}

.

General Inner Products

An inner product on a vector space

V

is a function

\langle \cdot, \cdot \rangle: V \times V \to \mathbb{R}

satisfying three axioms:

Inner Product Axioms

\begin{aligned} \text{Symmetry:} \quad & \langle \mathbf{u}, \mathbf{v} \rangle = \langle \mathbf{v}, \mathbf{u} \rangle \\ \text{Linearity:} \quad & \langle c\mathbf{u} + d\mathbf{w}, \mathbf{v} \rangle = c\langle \mathbf{u}, \mathbf{v} \rangle + d\langle \mathbf{w}, \mathbf{v} \rangle \\ \text{Positive definiteness:} \quad & \langle \mathbf{v}, \mathbf{v} \rangle > 0 \text{ for all } \mathbf{v} \neq \mathbf{0} \end{aligned}

Inner product axioms →

A vector space equipped with an inner product is called an inner product space. Every inner product induces a norm (

\|\mathbf{v}\| = \sqrt{\langle \mathbf{v}, \mathbf{v} \rangle}

), a distance (

d(\mathbf{u}, \mathbf{v}) = \|\mathbf{u} - \mathbf{v}\|

), and a notion of orthogonality (

\mathbf{u} \perp \mathbf{v}

iff

\langle \mathbf{u}, \mathbf{v} \rangle = 0

). The Cauchy-Schwarz inequality, the triangle inequality, and the Pythagorean theorem all hold in any inner product space.

The standard dot product on

\mathbb{R}^n

is one inner product. But the definition admits many others, each defining a different geometry on the same set of vectors.

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Orthogonality · general inner products

The three axioms

The dot product is one inner product among many. These three conditions are what any candidate must satisfy — and each one is responsible for a specific piece of geometry that would otherwise not exist.

06axioms

Unconditional1

ii

Linearity in the first argument§ 8

\langle c\mathbf{u} + d\mathbf{w}, \mathbf{v} \rangle = c\langle \mathbf{u}, \mathbf{v} \rangle + d\langle \mathbf{w}, \mathbf{v} \rangle

Combined with symmetry this gives linearity in both arguments — the form is bilinear, so an…

Unconditional

Requires: any scalars $c, d$

Combined with symmetry this gives linearity in both arguments — the form is bilinear, so an inner product distributes across linear combinations on either side. Only one argument needs stating; symmetry supplies the other.

Conditional5

i

Symmetry§ 8

\langle \mathbf{u}, \mathbf{v} \rangle = \langle \mathbf{v}, \mathbf{u} \rangle

Order of the arguments is irrelevant over ℝ.

Conditional

Holds when: real scalars
Fails when: complex scalars — becomes $\langle \mathbf{u}, \mathbf{v} \rangle = \overline{\langle \mathbf{v}, \mathbf{u} \rangle}$

Order of the arguments is irrelevant over $\mathbb{R}$ . Over $\mathbb{C}$ the axiom weakens to conjugate symmetry, and it has to — without the conjugate, positive definiteness below would fail, since $\langle i\mathbf{v}, i\mathbf{v} \rangle$ would come out negative.

iii

Positive definiteness§ 8

\langle \mathbf{v}, \mathbf{v} \rangle \geq 0

The axiom that supplies length.

Conditional

Holds when: equality iff $\mathbf{v} = \mathbf{0}$

The axiom that supplies length. It makes $\|\mathbf{v}\| = \sqrt{\langle \mathbf{v}, \mathbf{v} \rangle}$ real, non-negative, and zero only at the origin — see norm properties. Drop the strictness and nonzero vectors could have zero length, which breaks normalization and every distance argument built on it.

iv

Cauchy–Schwarz§ 6

|\langle \mathbf{u}, \mathbf{v} \rangle| \leq \|\mathbf{u}\| \, \|\mathbf{v}\|

Not an axiom — a consequence of the three above.

Derived

Holds when: equality iff $\mathbf{u}$ and $\mathbf{v}$ are parallel

Not an axiom — a consequence of the three above. It is what makes angle definable at all: without it, $\langle \mathbf{u}, \mathbf{v} \rangle / (\|\mathbf{u}\|\|\mathbf{v}\|)$ could fall outside $[-1, 1]$ and have no arccosine.

v

Triangle inequality§ 7

\|\mathbf{u} + \mathbf{v}\| \leq \|\mathbf{u}\| + \|\mathbf{v}\|

Follows from Cauchy–Schwarz by squaring both sides.

Derived

Holds when: equality iff $\mathbf{u}, \mathbf{v}$ point the same way

Follows from Cauchy–Schwarz by squaring both sides. This is what makes the induced norm a genuine norm and the induced distance a genuine metric — the three axioms give geometry, but only through this chain.

vi

Pythagorean theorem§ 10

\mathbf{u} \perp \mathbf{v} \Rightarrow \|\mathbf{u} + \mathbf{v}\|^2 = \|\mathbf{u}\|^2 + \|\mathbf{v}\|^2

Expand \|u+v\|^2 = u+v, u+v by bilinearity and the cross terms vanish.

Derived

Holds when: $\langle \mathbf{u}, \mathbf{v} \rangle = 0$

Expand $\|\mathbf{u}+\mathbf{v}\|^2 = \langle \mathbf{u}+\mathbf{v}, \mathbf{u}+\mathbf{v} \rangle$ by bilinearity and the cross terms vanish. The classical theorem in $\mathbb{R}^2$ turns out to be a statement about any inner product space — it holds for functions and matrices just as well.

Unconditional

Conditional

Symmetry§ 8Conditional

\langle \mathbf{u}, \mathbf{v} \rangle = \langle \mathbf{v}, \mathbf{u} \rangle

Holds when: real scalars
Fails when: complex scalars — becomes $\langle \mathbf{u}, \mathbf{v} \rangle = \overline{\langle \mathbf{v}, \mathbf{u} \rangle}$

Order of the arguments is irrelevant over

\mathbb{R}

. Over

\mathbb{C}

the axiom weakens to conjugate symmetry, and it has to — without the conjugate, positive definiteness below would fail, since

\langle i\mathbf{v}, i\mathbf{v} \rangle

would come out negative.

Read the full section

Linearity in the first argument§ 8Unconditional

\langle c\mathbf{u} + d\mathbf{w}, \mathbf{v} \rangle = c\langle \mathbf{u}, \mathbf{v} \rangle + d\langle \mathbf{w}, \mathbf{v} \rangle

Requires: any scalars $c, d$

Combined with symmetry this gives linearity in both arguments — the form is bilinear, so an inner product distributes across linear combinations on either side. Only one argument needs stating; symmetry supplies the other.

Read the full section

Positive definiteness§ 8Conditional

\langle \mathbf{v}, \mathbf{v} \rangle \geq 0

Holds when: equality iff $\mathbf{v} = \mathbf{0}$

The axiom that supplies length. It makes

\|\mathbf{v}\| = \sqrt{\langle \mathbf{v}, \mathbf{v} \rangle}

real, non-negative, and zero only at the origin — see norm properties. Drop the strictness and nonzero vectors could have zero length, which breaks normalization and every distance argument built on it.

Read the full section

Cauchy–Schwarz§ 6Derived

|\langle \mathbf{u}, \mathbf{v} \rangle| \leq \|\mathbf{u}\| \, \|\mathbf{v}\|

Holds when: equality iff $\mathbf{u}$ and $\mathbf{v}$ are parallel

Not an axiom — a consequence of the three above. It is what makes angle definable at all: without it,

\langle \mathbf{u}, \mathbf{v} \rangle / (\|\mathbf{u}\|\|\mathbf{v}\|)

could fall outside

[-1, 1]

and have no arccosine.

Read the full section

Triangle inequality§ 7Derived

\|\mathbf{u} + \mathbf{v}\| \leq \|\mathbf{u}\| + \|\mathbf{v}\|

Holds when: equality iff $\mathbf{u}, \mathbf{v}$ point the same way

Follows from Cauchy–Schwarz by squaring both sides. This is what makes the induced norm a genuine norm and the induced distance a genuine metric — the three axioms give geometry, but only through this chain.

Read the full section

Pythagorean theorem§ 10Derived

\mathbf{u} \perp \mathbf{v} \Rightarrow \|\mathbf{u} + \mathbf{v}\|^2 = \|\mathbf{u}\|^2 + \|\mathbf{v}\|^2

Holds when: $\langle \mathbf{u}, \mathbf{v} \rangle = 0$

Expand

\|\mathbf{u}+\mathbf{v}\|^2 = \langle \mathbf{u}+\mathbf{v}, \mathbf{u}+\mathbf{v} \rangle

by bilinearity and the cross terms vanish. The classical theorem in

\mathbb{R}^2

turns out to be a statement about any inner product space — it holds for functions and matrices just as well.

Read the full section

Every geometric notion on this page is derived from these three and nothing else. That is why the same machinery works on function spaces, matrix spaces and polynomial spaces — Gram–Schmidt and projection never look at what the vectors are, only at the axioms they obey.

The three axioms·/linear-algebra/orthogonality/inner-productLearn Math Class

Inner Product Notation

Notation

Inner Product Notation

The angle brackets' third and final job, the dots that mark empty argument slots, and the perpendicularity mark that grew a superscript.

The concrete dot — dot product notation;

\|\cdot\|

— norm notation; the other two angle-bracket jobs — span notation.

\langle \mathbf{u}, \mathbf{v} \rangle

The inner product of u and v

Angle brackets with a comma: two vectors in, one scalar out, by whatever rule the space has declared — the axioms of General Inner Products above. The dot product is the

\mathbb{R}^n

instance; the brackets are the abstraction that lets Fourier series and polynomial spaces borrow its geometry.

CasesThis closes the angle-bracket ledger: components in some calculus texts, generated subspaces in algebra, and inner products here — three jobs, and only context or a comma count separates them.

Also written

(\mathbf{u}, \mathbf{v})

— plain parentheses, the standard in Russian and much European literature; economical but colliding with points, tuples, and open intervals all at once.

Do not confuseDirac's bra-ket. Physics writes

\langle u \mid v \rangle

with a bar, not a comma — the same brackets split into freestanding halves

\langle u \mid

and

\mid v \rangle

; comma means mathematics, bar means quantum mechanics.

\langle \cdot, \cdot \rangle: V \times V \to \mathbb{R}

The inner product, as a function of two slots

The centered dots are placeholders: they mark empty argument slots, letting the operation be named without feeding it. The signature above — from the definition in General Inner Products — declares the machine: two vectors go in the slots, a real number comes out.

CasesThe same device names any operation slot-wise:

\|\cdot\|

for the norm,

d(\cdot, \cdot)

for distance,

f(\cdot)

for a function whose argument is momentarily none of your business.

Also writtenDashes or boxes in some texts (

\langle -, - \rangle

is common in category-flavoured writing) — same placeholder idea, different filler.

Do not confuseThe multiplication dot. Identical glyph, opposite role:

c \cdot \mathbf{v}

operates,

\langle \cdot, \cdot \rangle

leaves a blank — one is a verb, the other an empty chair.

\mathbf{u} \perp \mathbf{v}

·

W^{\perp}

u is orthogonal to v; W perp

The up-tack asserts

\langle \mathbf{u}, \mathbf{v} \rangle = 0

— geometry's perpendicularity sign, generalized to any inner product space. As a superscript it becomes an operator:

W^{\perp}

is the set of everything orthogonal to

W

, the orthogonal complement.

CasesThe mark scales across species:

\mathbf{u} \perp \mathbf{v}

between vectors,

\mathbf{v} \perp W

vector-to-subspace,

W^{\perp}

as a whole subspace — the four-subspace complement pairs are written exactly this way.

Also written“Orthogonal” spelled out where the symbol is unavailable;

\perp

itself is LaTeX \perp, distinct from the up-tack

\bot

of logic typography.

Do not confuseA superscript exponent or transpose.

W^{\perp}

joins

A^{T}

and

A^{-1}

in the crowded superscript corner — like them, it is a label naming an operation, and like them it reverses nothing about

W

itself.

All linear algebra symbols →↑ Orthogonality

Examples of Inner Products

The weighted inner product on

\mathbb{R}^n

is

\langle \mathbf{u}, \mathbf{v} \rangle = \mathbf{u}^T W \mathbf{v}

, where

W

is a symmetric positive definite matrix. This distorts the standard geometry — unit circles become ellipses, and "perpendicular" directions depend on

W

. When

W = I

, it reduces to the standard dot product.

On the polynomial space

\mathcal{P}_n

, the inner product

\langle p, q \rangle = \int_{-1}^{1} p(x)q(x)\,dx

defines orthogonality via integration. The polynomials

1

,

x

, and

\frac{1}{2}(3x^2 - 1)

are orthogonal under this product — these are the first three Legendre polynomials.

On the function space

C[0, 2\pi]

, the inner product

\langle f, g \rangle = \int_0^{2\pi} f(x)g(x)\,dx

makes sines and cosines orthogonal:

\int_0^{2\pi} \sin(mx)\cos(nx)\,dx = 0

for all integers

m, n

. This is the foundation of Fourier analysis.

The Frobenius inner product on matrices is

\langle A, B \rangle = \text{tr}(A^TB) = \sum_{ij} a_{ij}b_{ij}

, which treats matrices as vectors of

n^2

entries.

Each inner product defines its own geometry, but the linear algebra — projections, Gram-Schmidt, least squares — works identically in all of them.

The Pythagorean Theorem

If

\mathbf{u}

and

\mathbf{v}

are orthogonal (

\mathbf{u} \cdot \mathbf{v} = 0

), then

Pythagorean Theorem

\mathbf{u} \perp \mathbf{v} \implies \|\mathbf{u} + \mathbf{v}\|^2 = \|\mathbf{u}\|^2 + \|\mathbf{v}\|^2

Pythagorean theorem →

The proof is a one-line expansion:

\|\mathbf{u} + \mathbf{v}\|^2 = \mathbf{u} \cdot \mathbf{u} + 2\mathbf{u} \cdot \mathbf{v} + \mathbf{v} \cdot \mathbf{v} = \|\mathbf{u}\|^2 + 0 + \|\mathbf{v}\|^2

.

The theorem extends to any number of mutually orthogonal vectors: if

\mathbf{v}_1, \dots, \mathbf{v}_k

are pairwise orthogonal, then

\|\mathbf{v}_1 + \mathbf{v}_2 + \cdots + \mathbf{v}_k\|^2 = \|\mathbf{v}_1\|^2 + \|\mathbf{v}_2\|^2 + \cdots + \|\mathbf{v}_k\|^2

All cross terms vanish because every pair has dot product zero. This is not a special property of

\mathbb{R}^2

or

\mathbb{R}^3

— it holds in any inner product space. The Pythagorean theorem is a direct consequence of the inner product axioms, and it is the reason that orthogonal decompositions are so computationally clean: lengths decompose into independent, additive contributions from each perpendicular direction.

Relation	Statement	When equality holds	What it provides
Cauchy–Schwarz inequality	\|⟨u, v⟩\| ≤ ‖u‖ · ‖v‖	u and v are parallel (one is a scalar multiple of the other)	legitimacy of the angle formula: −1 ≤ cos θ ≤ 1; bounds on inner products in any inner product space
Triangle inequality	‖u + v‖ ≤ ‖u‖ + ‖v‖	u and v point in the same direction (one is a non-negative multiple of the other)	the induced distance d(u, v) = ‖u − v‖ satisfies the metric axioms
Pythagorean theorem	u ⊥ v ⟹ ‖u + v‖² = ‖u‖² + ‖v‖²	always (the statement is an equality, not an inequality, valid whenever u ⊥ v)	cross terms vanish; lengths add additively across orthogonal directions, making orthogonal decompositions clean

Summary: What an Inner Product Induces

Every quantity built in this page — norm, distance, angle, the orthogonality criterion, and the projection formula on the projections page — is constructed from a single inner product through a small family of formulas. The table below collects each induced quantity alongside its general definition in any inner product space, the concrete dot-product form in ℝⁿ, and a pointer back to the section that introduced it.

Quantity induced by ⟨·, ·⟩	General definition	Standard dot-product instance in ℝⁿ	Where introduced
Norm (length) of a vector	‖v‖ = √⟨v, v⟩	√(v · v) = √(v₁² + … + v_n²)	obj3 — Length
Unit vector / normalization	v̂ = v / ‖v‖	v / √(v · v)	obj3 — Length
Distance between vectors	d(u, v) = ‖u − v‖	√Σ_i (u_i − v_i)²	obj4 — Distance
Angle between vectors	cos θ = ⟨u, v⟩ / (‖u‖ · ‖v‖)	(u · v) / (‖u‖ · ‖v‖)	obj5 — Angle Between Vectors
Orthogonality	u ⊥ v ⟺ ⟨u, v⟩ = 0	u · v = 0	obj5 — Angle Between Vectors (special case θ = π/2)
Projection of u onto a nonzero v	proj_v(u) = (⟨u, v⟩ / ⟨v, v⟩) · v	((u · v) / (v · v)) · v	covered on the projections page

Inner Product FAQ

Is $\langle u \mid v \rangle$ the same as $\langle u, v \rangle$ ?

+

Same idea, different dialect. A comma marks the mathematician's inner product; a vertical bar marks Dirac's bra-ket from quantum mechanics. The bar version is built to split into freestanding halves that carry meaning on their own, which the comma form never does. Comma means mathematics, bar means physics.Read more →

What do the dots mean in $\langle \cdot, \cdot \rangle$ ?

+

They are empty slots, marking where arguments would go when you want to name the operation without applying it. The same device gives

\|\cdot\|

for a norm and

d(\cdot, \cdot)

for a distance. Note the glyph is identical to the multiplication dot but plays the opposite role: one is a verb, the other an empty chair.Read more →

Is the $\perp$ in $W^{\perp}$ an exponent?

+

No, it is a label, joining

A^{T}

and

A^{-1}

in the crowded superscript corner where none of the marks raise anything to a power. The symbol also scales across species:

\mathbf{u} \perp \mathbf{v}

relates two vectors,

\mathbf{v} \perp W

a vector to a subspace, and

W^{\perp}

names a whole subspace.Read more →

Inner Product

Measuring Length, Angle, and Distance

The Dot Product

Properties of the Dot Product

Length

Distance

Angle Between Vectors

The Cauchy-Schwarz Inequality

The Triangle Inequality

General Inner Products

The three axioms

Inner Product Notation

Inner Product Notation

Examples of Inner Products

The Pythagorean Theorem

Summary: What an Inner Product Induces

Inner Product FAQ

Is ⟨u∣v⟩\langle u \mid v \rangle⟨u∣v⟩ the same as ⟨u,v⟩\langle u, v \rangle⟨u,v⟩?

What do the dots mean in ⟨⋅,⋅⟩\langle \cdot, \cdot \rangle⟨⋅,⋅⟩?

Is the ⊥\perp⊥ in W⊥W^{\perp}W⊥ an exponent?

Is $\langle u \mid v \rangle$ the same as $\langle u, v \rangle$ ?

What do the dots mean in $\langle \cdot, \cdot \rangle$ ?

Is the $\perp$ in $W^{\perp}$ an exponent?