What is the dot product of two vectors?

The dot product of u and v in Rⁿ is u·v = u₁v₁ + u₂v₂ + ⋯ + uₙvₙ, producing a scalar. It can also be written as the matrix product uᵀv. The dot product defines lengths, angles, distances, and perpendicularity in finite-dimensional linear algebra.

How do you find the angle between two vectors?

The angle θ between nonzero vectors u and v satisfies cos θ = (u·v)/(‖u‖‖v‖). The Cauchy-Schwarz inequality guarantees this ratio lies between −1 and 1. When the dot product is zero, the angle is 90° and the vectors are orthogonal.

What is the Cauchy-Schwarz inequality?

The Cauchy-Schwarz inequality states |u·v| ≤ ‖u‖‖v‖ for all vectors u and v. Equality holds if and only if the vectors are parallel. It ensures the angle formula always produces a valid cosine value and is fundamental to proving the triangle inequality.

What is an inner product space?

An inner product space is a vector space equipped with an inner product — a function satisfying symmetry, linearity, and positive definiteness. Every inner product induces a norm, a distance, and a notion of orthogonality. The standard dot product on Rⁿ is one example; integrals on function spaces and weighted products are others.

What is the Pythagorean theorem for vectors?

If vectors u and v are orthogonal (u·v = 0), then ‖u + v‖² = ‖u‖² + ‖v‖². This extends to any number of mutually orthogonal vectors — all cross terms vanish. It holds in any inner product space and is why orthogonal decompositions produce cleanly additive length contributions.

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.

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.

Distance

The distance between two vectors

\mathbf{u}

and

\mathbf{v}

is the length of their difference:

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

This is 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

:

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

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

:

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

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:

Symmetry:

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

.

Linearity in the first argument:

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

.

Positive definiteness:

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

for all

\mathbf{v} \neq \mathbf{0}

.

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.

Axiom	General form ⟨·, ·⟩	Dot-product instance in ℝⁿ	What it ensures
Symmetry	⟨u, v⟩ = ⟨v, u⟩	u · v = v · u	order of the two arguments is irrelevant
Linearity (in the first argument)	⟨c u + d w, v⟩ = c⟨u, v⟩ + d⟨w, v⟩	(c u + d w) · v = c (u · v) + d (w · v)	combined with symmetry, ⟨·, ·⟩ is bilinear; lets inner products distribute across linear combinations
Positive definiteness	⟨v, v⟩ ≥ 0, with equality iff v = 0	v · v = v₁² + … + v_n² ≥ 0, zero only when v = 0	the induced norm ‖v‖ = √⟨v, v⟩ is real, non-negative, and vanishes only at the origin

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

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

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

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

Examples of Inner Products

The Pythagorean Theorem

Summary: What an Inner Product Induces