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Euclidean vector

A Euclidean vector, also known as a geometric or spatial vector, is a fundamental mathematical object in Euclidean space that possesses both magnitude (length) and direction, typically represented as a directed line segment from an initial point (tail) to a terminal point (head) or as an ordered tuple of real numbers corresponding to coordinates in \mathbb{R}^n.^[1] These vectors form the basis of vector spaces over the real numbers, where the Euclidean structure is defined by an inner product, such as the dot product, which induces a norm measuring length and orthogonality between vectors.^[2]^[3] Euclidean vectors support key operations including addition, which combines two vectors tail-to-head to form a resultant vector, and scalar multiplication, which scales the magnitude and potentially reverses direction based on the scalar's sign; these operations satisfy properties like commutativity, associativity, and distributivity, making the set a vector space.^[4]^[5] The magnitude, or Euclidean norm, of a vector \mathbf{v} = (v_1, v_2, \dots, v_n) is given by \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \dots + v_n^2}, derived from the dot product \mathbf{v} \cdot \mathbf{v}, enabling concepts like unit vectors (norm 1) and projections.^[6]^[7] Additionally, the dot product \mathbf{u} \cdot \mathbf{v} = \sum u_i v_i quantifies the angle between vectors, with orthogonality when it equals zero, and supports decomposition into orthogonal components.^[8] In applications, Euclidean vectors are essential in physics for representing forces, velocities, and displacements, in computer graphics for transformations and rendering, and in data science for multidimensional analysis, underpinning fields from classical mechanics to machine learning algorithms.^[9] Their finite-dimensional nature distinguishes them from more abstract vectors in infinite-dimensional spaces, while generalizations extend to non-Euclidean geometries.^[10]

Overview

Definition and basic characteristics

A Euclidean vector is a mathematical object that represents a directed line segment in n-dimensional Euclidean space, characterized by both magnitude (or length) and direction.^[1] This geometric interpretation allows vectors to model displacements or quantities where orientation matters, such as force or velocity in physical contexts, though the core concept remains algebraic and independent of specific applications.^[11] Unlike scalars, which are quantities defined solely by magnitude (e.g., temperature or mass), vectors require both a numerical size and a directional specification to fully describe them.^[12] In Euclidean space, vectors exhibit key characteristics that distinguish their behavior. They can be classified as free vectors, whose position is irrelevant and can be translated without altering their properties, or bound vectors, which are fixed to a specific point of application.^[13] Free vectors emphasize the equivalence of parallel segments with equal length and direction, enabling their use in abstract algebraic manipulations.^[14] Visually, vectors are often depicted as arrows, where the arrow's length corresponds to magnitude and its orientation to direction, providing an intuitive aid grounded in basic Euclidean geometry.^[5] Common notation for vectors includes boldface letters, such as \mathbf{v}, or an arrow overhead, like \vec{v}, to differentiate them from scalar variables.^[15] In component form, a vector in n-dimensional space is expressed as \mathbf{v} = (v_1, v_2, \dots, v_n), where each v_i is a real number representing the projection along the i-th coordinate axis in a chosen basis.^[16] This representation assumes familiarity with the standard Euclidean metric on \mathbb{R}^n, facilitating both geometric intuition and computational handling.^[17]

Examples in one dimension

In one dimension, a Euclidean vector is fundamentally a signed scalar quantity along a straight line, where the positive sign denotes one direction (e.g., to the right or east) and the negative sign denotes the opposite direction (e.g., to the left or west). This representation captures both magnitude and direction on a single axis, forming the simplest case of a vector space \mathbb{R}^1.^[18] A classic example is displacement on a number line. Suppose an object moves from position 0 to +5 units along the x-axis; this displacement is the one-dimensional vector \mathbf{d} = +5, indicating a shift eastward. Conversely, a movement from 0 to -3 units represents \mathbf{d} = -3, signifying a westward shift. These signed values encode the net change in position, distinguishing them from mere distances, which are always positive scalars.^[18]^[19] Another illustrative case is velocity as a one-dimensional vector. For an object traveling at 10 m/s to the east, the velocity vector is \mathbf{v} = +10 m/s, while 4 m/s to the west is \mathbf{v} = -4 m/s. This signed scalar conveys both speed and directional sense along the line, essential for describing linear motion.^[20] Visually, one-dimensional vectors are depicted as directed arrows on a number line: a positive vector points rightward with length proportional to its value, while a negative one points leftward. The magnitude, or length, of such a vector \mathbf{v} = v is the absolute value |\mathbf{v}| = |v|, which ignores direction and yields a non-negative scalar.^[18]^[21] This foundational one-dimensional framework extends seamlessly to higher dimensions, where vectors gain additional components perpendicular to the original axis, building toward the full structure of Euclidean space.^[21]

Role in physics and engineering

In physics, Euclidean vectors are essential for representing quantities that possess both magnitude and direction, distinguishing them from scalars, which have only magnitude. For instance, displacement, velocity, acceleration, and force are modeled as vectors because their effects depend on how they point in space, whereas properties like mass, temperature, or speed are scalars unaffected by direction.^[22] This vectorial nature allows physicists to describe the motion of objects accurately; for example, an object's velocity vector captures not just its speed but also its trajectory, enabling predictions of future positions under various influences.^[23] Similarly, forces acting on a body, such as gravitational or frictional forces, are treated as vectors to account for their directional push or pull, which is crucial for analyzing interactions in systems like planetary orbits or collisions.^[24] In engineering, Euclidean vectors bridge theoretical mechanics to practical design, particularly in statics and dynamics. In statics, vectors model forces and moments in structures at rest, such as the equilibrium of beams or trusses under loads, where resolving multiple forces into components ensures stability without net motion.^[25] Dynamics extends this to moving systems, using vectors for acceleration and momentum to simulate vehicle trajectories or robotic arms, allowing engineers to optimize performance and safety.^[26] A key utility is the decomposition of complex motions or forces into orthogonal components, simplifying analysis of multifaceted problems like aircraft flight paths influenced by wind and thrust.^[1] Beyond mechanics, vectors appear in electrical engineering through phasors, which represent sinusoidal voltages and currents as rotating vectors in the complex plane to analyze alternating current (AC) circuits. Phasors facilitate the summation of waveforms with different phases, akin to vector addition, enabling efficient computation of circuit impedance and power without time-domain integration.^[27] This approach is foundational in designing power systems and signal processors, where directional phase relationships determine overall behavior.^[28]

History

Early geometric intuitions

The earliest intuitions of vector-like concepts emerged in ancient Greek geometry, where directed quantities were represented through visual and geometric means rather than algebraic notation. In Euclid's Elements (c. 300 BCE), the parallel postulate and related propositions implicitly addressed direction by preserving the orientation of lines under translation, allowing for the conceptual treatment of displacements as having both magnitude and sense, though without a unified vector framework. Similarly, in optics, Euclid's Optics employed ray diagrams to depict light propagation as straight lines emanating from the eye, illustrating directional paths that anticipated modern vector representations of propagation, as seen in propositions describing rays falling on surfaces and their reflections. These geometric constructions, including the parallelogram rule for composing velocities attributed to various ancient authors and even Pseudo-Aristotle's Mechanical Problems, treated motions as directed segments that could be added head-to-tail, forming a foundational intuitive grasp of vector addition without formal symbolism.^[29] In the 17th century, these geometric ideas gained prominence in physics through the work of Isaac Newton, who conceptualized forces as directed quantities in his Philosophiæ Naturalis Principia Mathematica (1687). Newton explicitly invoked the parallelogram law to compose forces, stating in Corollary I to the Laws of Motion that if two forces act simultaneously on a body, the resultant is determined by completing a parallelogram with the forces as adjacent sides, yielding the diagonal as the effective directed impetus.^[30] This approach built on earlier intuitions but integrated them into dynamical analysis, where forces were visualized as arrows indicating both intensity and direction, essential for explaining curvilinear motion under central attractions.^[29] Gottfried Wilhelm Leibniz contributed parallel developments in his writings on motion and calculus, introducing the "characteristic triangle" to geometrically represent infinitesimal changes in position and velocity, aligning with the parallelogram law for composing directed speeds and forces as segments in a plane.^[31] Key advancements in force composition occurred in the 1670s among British and continental natural philosophers. Christopher Wren, Robert Hooke, and Christiaan Huygens independently explored the synthesis of directed forces in planetary motion, with Wren and Hooke hypothesizing an inverse-square law for gravitational attraction by combining Huygens's 1673 expression for centrifugal force with Kepler's harmonic law, using geometric diagrams to resolve composite effects.^[32] These efforts, often depicted as arrows or line segments denoting direction and magnitude, exemplified vectors as "geometric arrows" devoid of algebraic machinery, relying instead on Euclidean constructions to model physical interactions like orbital deviations.^[33] In contrast, the later 19th-century formalizations by J. Willard Gibbs and Oliver Heaviside introduced rigorous algebraic vector systems, marking a shift from these intuitive geometric precedents.^[34]

Formal mathematical development

The formal mathematical development of Euclidean vectors began in the mid-19th century, building on earlier geometric intuitions to establish an algebraic framework for vector operations and structures. In 1843, William Rowan Hamilton introduced quaternions as a four-dimensional extension of complex numbers, which provided a multiplicative structure that influenced the emergence of vector analysis by separating scalar and vector components in three dimensions. This innovation laid groundwork for treating vectors as algebraic entities rather than purely geometric objects. Concurrently, in 1844, Hermann Grassmann published his Ausdehnungslehre (Theory of Extension), which developed the concept of multivectors—higher-grade extensions combining vectors and their products—offering a comprehensive algebraic system for linear combinations and outer products in multidimensional spaces.^[35] By the 1880s, the work of Josiah Willard Gibbs and Oliver Heaviside standardized vector analysis through treatises that emphasized practical notation for vectors in physics, including the dot and cross products, while discarding the quaternion's scalar components to focus on three-dimensional Euclidean vectors.^[36] Their independent developments, disseminated via Gibbs's Yale lectures and Heaviside's electromagnetic papers, established a concise symbolic language that became the foundation for modern vector calculus. This period marked a pivotal shift from synthetic geometry—relying on axiomatic constructions without coordinates—to analytic methods using algebraic manipulations and coordinate representations, enabling rigorous proofs and computations in higher dimensions.^[37] In the 20th century, Hermann Minkowski extended Euclidean vectors to four-dimensional space-time in his 1908 formulation, integrating time as a fourth coordinate to unify space and time under Lorentz transformations in special relativity, thus broadening vector applications beyond classical mechanics.^[38] Parallel to this, the axiomatization of vector spaces in linear algebra progressed with Giuseppe Peano's 1888 definition of abstract vector spaces over fields, providing a general framework independent of dimension, which was further refined in David Hilbert's work on infinite-dimensional spaces. Hilbert spaces, introduced around 1906–1910, axiomatized Euclidean vectors as complete inner product spaces, encompassing both finite- and infinite-dimensional cases and underpinning functional analysis. A key educational milestone was Charles Jasper Joly's 1901 edition of Hamilton's Elements of Quaternions, which served as an accessible primer linking quaternion-based vectors to contemporary algebraic developments.^[39]

Representations

Cartesian coordinates

In Euclidean space, vectors are commonly represented using Cartesian coordinates, which assign numerical values to their positions relative to a fixed origin and orthogonal axes. This system allows a vector to be expressed as an ordered tuple of components, corresponding to displacements along each coordinate direction. For instance, in three-dimensional space, a vector \mathbf{v} is written as \mathbf{v} = (x, y, z), where x, y, and z are the scalar components./10:_Vectors/10.01:_Introduction_to_Cartesian_Coordinates_in_Space) Equivalently, this representation can be formulated as a linear combination of basis vectors: \mathbf{v} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}, where \mathbf{i}, \mathbf{j}, and \mathbf{k} are the standard unit basis vectors pointing along the positive x-, y-, and z-axes, respectively. In n-dimensional Euclidean space, the general form is \mathbf{v} = \sum_{i=1}^n v_i \mathbf{e}_i, with \mathbf{e}_i denoting the i-th basis vector. These basis vectors form an orthonormal set, meaning each has unit magnitude (\|\mathbf{e}_i\| = 1) and they are pairwise orthogonal (\mathbf{e}_i \cdot \mathbf{e}_j = 0 for i \neq j). The orthonormality condition is compactly expressed as \mathbf{e}_i \cdot \mathbf{e}_j = \delta_{ij}, where \delta_{ij} is the Kronecker delta (equal to 1 if i = j and 0 otherwise).^[40]^[41] The components of a vector in this basis are extracted through projection onto the corresponding basis vector. Specifically, the i-th component is given by v_i = \mathbf{v} \cdot \mathbf{e}_i, leveraging the orthonormality to isolate the scalar projection without interference from other directions.^[40]^[41] This Cartesian representation is foundational because it aligns directly with the geometric structure of Euclidean space, enabling straightforward algebraic manipulations and visualizations. Its orthogonality simplifies computations involving distances, angles, and transformations, making it the preferred system for most analytical work in physics and engineering.^[42]^[43]

Decomposition into components

In Euclidean space, a vector \mathbf{v} is decomposed into its components relative to an orthonormal basis, expressing it as a sum of scalar multiples of the basis vectors. In three dimensions, using the standard Cartesian basis \mathbf{i}, \mathbf{j}, and \mathbf{k} (unit vectors along the coordinate axes), this resolution takes the form

\mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k},

where v_x, v_y, and v_z are the scalar components representing the directed distances along each axis.^[44] This decomposition is unique in an orthonormal basis and facilitates coordinate-based calculations.^[24] The scalar components are obtained via the dot product with the basis vectors: v_x = \mathbf{v} \cdot \mathbf{i}, v_y = \mathbf{v} \cdot \mathbf{j}, and v_z = \mathbf{v} \cdot \mathbf{k}, where the dot product \mathbf{a} \cdot \mathbf{b} = \sum a_k b_k extracts the projection in the basis direction.^[45] To derive this, start with the decomposition equation and compute the dot product with \mathbf{i}:

\mathbf{v} \cdot \mathbf{i} = (v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k}) \cdot \mathbf{i} = v_x (\mathbf{i} \cdot \mathbf{i}) + v_y (\mathbf{j} \cdot \mathbf{i}) + v_z (\mathbf{k} \cdot \mathbf{i}).

Orthonormality implies \mathbf{i} \cdot \mathbf{i} = 1 and \mathbf{j} \cdot \mathbf{i} = \mathbf{k} \cdot \mathbf{i} = 0, so the equation reduces to v_x = \mathbf{v} \cdot \mathbf{i}. The same holds for the other components by symmetry.^[46] Geometrically, each component v_x is the signed scalar projection of \mathbf{v} onto the x-axis, equal to |\mathbf{v}| \cos \theta_x, where \theta_x is the angle between \mathbf{v} and \mathbf{i}.^[45] In two dimensions, the decomposition aligns with the parallelogram rule: the component vectors v_x \mathbf{i} and v_y \mathbf{j} form adjacent sides of a parallelogram, with \mathbf{v} as the resultant diagonal.^[44] This projection-based interpretation underscores the dot product's role in quantifying directional alignment.^[24] This component decomposition simplifies analysis in Euclidean space by converting vector equations into independent scalar ones per direction, essential for solving problems in physics and engineering. While non-orthogonal bases require more involved projections, the orthogonal Euclidean case yields straightforward, computationally efficient resolutions.^[24]

Properties

Equality and parallelism

In Euclidean space, two vectors \mathbf{u} and \mathbf{v} are equal if and only if they have the same dimension and their corresponding components are identical, that is, u_i = v_i for every index i = 1, 2, \dots, n. This component-wise equality ensures that the vectors represent the same directed quantity in the space.^[4] Geometrically, equal vectors are represented by arrows that have identical length (magnitude) and direction, though they may originate from different points in space.^[47] This interpretation aligns with the free vector concept, where position is irrelevant, and only the displacement matters.^[47] Two vectors \mathbf{u} and \mathbf{v} are parallel if one is a scalar multiple of the other, \mathbf{u} = k \mathbf{v} for some nonzero scalar k \in \mathbb{R}. If k > 0, the vectors point in the same direction; if k < 0, they are antiparallel, pointing in opposite directions. Geometrically, parallel vectors point in the same or opposite directions and can be depicted as arrows that are parallel, regardless of their positions in space, though they may not share the same magnitude.^[48] In three-dimensional Euclidean space, two nonzero vectors are parallel if and only if their cross product is the zero vector, \mathbf{u} \times \mathbf{v} = \mathbf{0}. The cross product \mathbf{u} \times \mathbf{v} produces a vector perpendicular to the plane spanned by \mathbf{u} and \mathbf{v}, with magnitude |\mathbf{u}| |\mathbf{v}| \sin \theta, where \theta is the angle between them; this magnitude is zero precisely when \theta = 0^\circ or $180^\circ, confirming parallelism or antiparallelism.^[49]

Magnitude and unit vectors

The magnitude, or Euclidean norm, of a vector \mathbf{v} = (v_1, v_2, \dots, v_n) in \mathbb{R}^n is defined as \|\mathbf{v}\| = \sqrt{\sum_{i=1}^n v_i^2}.^[50]^[6] This definition arises from the Pythagorean theorem applied iteratively to the components of the vector. In two dimensions, for \mathbf{v} = (v_1, v_2), the magnitude is the length of the hypotenuse of a right triangle with legs |v_1| and |v_2|, given by \sqrt{v_1^2 + v_2^2}.^[51] Extending to higher dimensions, the norm is computed by first finding the magnitude in the subspace of the first n-1 components, say \sqrt{\sum_{i=1}^{n-1} v_i^2}, and then applying the Pythagorean theorem again with the nth component: \|\mathbf{v}\| = \sqrt{\left( \sqrt{\sum_{i=1}^{n-1} v_i^2} \right)^2 + v_n^2} = \sqrt{\sum_{i=1}^n v_i^2}.^[52]^[53] Equivalently, the magnitude can be expressed using the dot product as \|\mathbf{v}\| = \sqrt{\mathbf{v} \cdot \mathbf{v}}, where the dot product provides the basis for this norm in Euclidean space.^[54]^[55] A unit vector, or normalized vector, is obtained by dividing a nonzero vector by its magnitude: \hat{\mathbf{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}, which has magnitude 1 and preserves the direction of \mathbf{v}.^[56]^[57] This normalization process isolates the directional aspect of the vector, making unit vectors useful for indicating orientation without scaling information.^[58] The zero vector \mathbf{0} = (0, 0, \dots, 0) has magnitude \|\mathbf{0}\| = 0 and no defined direction, as it represents a point with no displacement.^[59] It serves as the additive identity in the vector space, satisfying \mathbf{0} + \mathbf{v} = \mathbf{v} and \mathbf{v} + \mathbf{0} = \mathbf{v} for any vector \mathbf{v}, and is unique in this role.^[60] Unlike nonzero vectors, the zero vector cannot be normalized, as division by its magnitude is undefined.^[61]

Operations

Addition, subtraction, and scalar multiplication

In Euclidean vector spaces, addition of two vectors \mathbf{u} and \mathbf{v} is defined component-wise in Cartesian coordinates: if \mathbf{u} = (u_1, u_2, \dots, u_n) and \mathbf{v} = (v_1, v_2, \dots, v_n), then \mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, \dots, u_n + v_n)./04%3A_R/4.02%3A_Vector_Algebra) This operation satisfies commutativity, \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}, and associativity, (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}), for any vectors \mathbf{u}, \mathbf{v}, \mathbf{w}.^[5] Geometrically, vector addition follows the parallelogram law: the sum \mathbf{u} + \mathbf{v} is the diagonal of the parallelogram formed by \mathbf{u} and \mathbf{v} as adjacent sides.^[9] Equivalently, the triangle rule constructs the sum by attaching the tail of \mathbf{v} to the head of \mathbf{u}, with \mathbf{u} + \mathbf{v} extending from the tail of \mathbf{u} to the head of \mathbf{v}.^[62] Vector subtraction is derived from addition via the additive inverse: \mathbf{u} - \mathbf{v} = \mathbf{u} + (-\mathbf{v}), where the opposite vector -\mathbf{v} points in the reverse direction with the same magnitude as \mathbf{v}./04%3A_R/4.02%3A_Vector_Algebra) Component-wise, -\mathbf{v} = (-v_1, -v_2, \dots, -v_n), obtained by multiplying \mathbf{v} by the scalar -1.^[5] This definition preserves the algebraic structure, allowing subtraction to be visualized as addition of the reversed vector using the parallelogram or triangle rules.^[62] Scalar multiplication scales a vector \mathbf{v} by a real number k: k\mathbf{v} = (k v_1, k v_2, \dots, k v_n), which stretches or compresses the vector while preserving its direction (reversing it if k < 0)./04%3A_R/4.02%3A_Vector_Algebra) Key properties include distributivity over vector addition, k(\mathbf{u} + \mathbf{v}) = k\mathbf{u} + k\mathbf{v}, and over scalar addition, (k + m)\mathbf{v} = k\mathbf{v} + m\mathbf{v}, along with associativity, k(m\mathbf{v}) = (km)\mathbf{v}.^[5] These operations collectively form the foundation of the vector space axioms for Euclidean spaces, enabling linear combinations such as a\mathbf{u} + b\mathbf{v}.^[63]

Dot product and orthogonality

The dot product, also known as the scalar product or inner product, of two Euclidean vectors \mathbf{u} = (u_1, u_2, \dots, u_n) and \mathbf{v} = (v_1, v_2, \dots, v_n) in \mathbb{R}^n is defined algebraically as the sum of the products of their corresponding components:

\mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^n u_i v_i.

This operation yields a scalar value and is fundamental to the geometry of Euclidean space./12%3A_Vectors_in_Space/12.03%3A_The_Dot_Product)^[64] Geometrically, the dot product is expressed in terms of the magnitudes of the vectors and the angle \theta between them (where $0 \leq \theta \leq \pi):

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

with the magnitude \|\mathbf{u}\| defined as the square root of \mathbf{u} \cdot \mathbf{u}. This formulation arises from considering the projection of one vector onto the other, scaled by the magnitude of the second vector.^[65]^[66] To derive the equivalence between the algebraic and geometric definitions, start with the law of cosines in the triangle formed by \mathbf{u}, \mathbf{v}, and \mathbf{u} - \mathbf{v}. The squared length of \mathbf{u} - \mathbf{v} is \|\mathbf{u} - \mathbf{v}\|^2 = \|\mathbf{u}\|^2 + \|\mathbf{v}\|^2 - 2 \|\mathbf{u}\| \|\mathbf{v}\| \cos \theta. Expanding algebraically gives:

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

Equating the two expressions yields \mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \theta, confirming the definitions are consistent for Euclidean vectors.^[67]^[64]^[68] The dot product satisfies several key properties that underpin its utility in vector analysis. It is commutative, meaning \mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}, and bilinear, linear in each argument separately: (\mathbf{u} + \mathbf{w}) \cdot \mathbf{v} = \mathbf{u} \cdot \mathbf{v} + \mathbf{w} \cdot \mathbf{v} and (c \mathbf{u}) \cdot \mathbf{v} = c (\mathbf{u} \cdot \mathbf{v}) for any scalar c. Additionally, it is positive definite: \mathbf{u} \cdot \mathbf{u} \geq 0, with equality if and only if \mathbf{u} = \mathbf{0}. These properties make the dot product an inner product on the Euclidean space \mathbb{R}^n.^[66]^[64]/12%3A_Vectors_in_Space/12.03%3A_The_Dot_Product) Two nonzero vectors \mathbf{u} and \mathbf{v} are orthogonal (perpendicular) if and only if their dot product is zero: \mathbf{u} \cdot \mathbf{v} = 0. This condition follows directly from the geometric definition, as \cos \theta = 0 when \theta = \pi/2. Orthogonality is a cornerstone of Euclidean geometry, enabling decompositions into perpendicular components and forming the basis for orthogonal bases in vector spaces.^[66]^[64]/12%3A_Vectors_in_Space/12.03%3A_The_Dot_Product) In applications, the dot product determines the angle between two vectors via \theta = \arccos \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} \right), providing a measure of their directional similarity independent of magnitude. It also computes the scalar projection of \mathbf{u} onto \mathbf{v}, given by \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|}, and the vector projection:

\proj_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \mathbf{v}.

This projection represents the component of \mathbf{u} in the direction of \mathbf{v}, essential for resolving vectors into orthogonal parts.^[65]^[66]^[64]

Cross product and triple products

In three-dimensional Euclidean space, the cross product of two vectors \mathbf{u} and \mathbf{v} is defined as a vector \mathbf{u} \times \mathbf{v} that is perpendicular to both \mathbf{u} and \mathbf{v}, with a magnitude equal to the area of the parallelogram they span.^[49] The direction of \mathbf{u} \times \mathbf{v} follows the right-hand rule: pointing the fingers of the right hand from \mathbf{u} toward \mathbf{v} aligns the thumb with the direction of the resulting vector.^[69] The magnitude satisfies \|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\| \|\mathbf{v}\| \sin \theta, where \theta is the angle between \mathbf{u} and \mathbf{v}.^[69] In Cartesian coordinates, with \mathbf{u} = (u_1, u_2, u_3) and \mathbf{v} = (v_1, v_2, v_3), the cross product is given by the determinant formula:

\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix} = (u_2 v_3 - u_3 v_2) \mathbf{i} - (u_1 v_3 - u_3 v_1) \mathbf{j} + (u_1 v_2 - u_2 v_1) \mathbf{k}.

^[70] This yields the component form \mathbf{u} \times \mathbf{v} = (u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_1 v_2 - u_2 v_1).^[69] Key properties of the cross product include anticommutativity, \mathbf{u} \times \mathbf{v} = -\mathbf{v} \times \mathbf{u}, which follows from the determinant representation and implies that the operation reverses direction when operands are swapped.^[69] The result is always perpendicular to both input vectors, as their dot products with \mathbf{u} \times \mathbf{v} vanish: \mathbf{u} \cdot (\mathbf{u} \times \mathbf{v}) = 0 and \mathbf{v} \cdot (\mathbf{u} \times \mathbf{v}) = 0.^[69] Additionally, \mathbf{u} \times \mathbf{v} = \mathbf{0} if and only if \mathbf{u} and \mathbf{v} are parallel (or at least one is the zero vector), since \sin \theta = 0 in that case.^[69] The scalar triple product extends the cross product to three vectors \mathbf{u}, \mathbf{v}, and \mathbf{w}, defined as \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}).^[71] This equals the determinant of the 3×3 matrix whose columns (or rows) are the components of \mathbf{u}, \mathbf{v}, and \mathbf{w}:

\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = \det \begin{pmatrix} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{pmatrix}.

^[71] Geometrically, the absolute value |\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})| represents the signed volume of the parallelepiped spanned by the three vectors, with the sign indicating orientation relative to the standard right-handed basis.^[71] The scalar triple product is invariant under cyclic permutation of the vectors but changes sign under odd permutations.^[69]

Applications in Physics

Kinematics: position, velocity, acceleration

In kinematics, the position of a particle in Euclidean space is described by the position vector \mathbf{r}(t), which extends from a chosen origin to the particle's location at time t. This vector encapsulates the full spatial coordinates of the particle relative to the origin, allowing for a complete geometric representation of its placement in three-dimensional space.^[72]^[73] The velocity vector \mathbf{v}(t) quantifies the particle's rate of change of position. The average velocity over a time interval \Delta t is defined as the change in position vector divided by the time elapsed, \mathbf{v}_\text{avg} = \frac{\Delta \mathbf{r}}{\Delta t}, which represents the straight-line displacement per unit time.^[74]^[75] In the limit as \Delta t approaches zero, this yields the instantaneous velocity, \mathbf{v}(t) = \frac{d\mathbf{r}}{dt}, the first time derivative of the position vector, capturing the particle's velocity at a precise instant.^[76]^[77] For constant velocity, the displacement simplifies to \Delta \mathbf{r} = \mathbf{v} \Delta t.^[78] The acceleration vector \mathbf{a}(t) measures the rate of change of velocity, defined as the time derivative \mathbf{a}(t) = \frac{d\mathbf{v}}{dt} = \frac{d^2 \mathbf{r}}{dt^2}. This second derivative of the position vector describes how the particle's motion evolves, with its direction and magnitude indicating changes in speed or direction.^[79]^[77] In curvilinear motion, acceleration decomposes into tangential and normal components: the tangential component a_T = \frac{dv}{dt} aligns with the velocity and affects speed, while the normal component a_N = \frac{v^2}{\rho} (where \rho is the radius of curvature) points toward the center of curvature and governs directional change.^[80]^[81] Vector differentiation in kinematics follows standard rules, such as the product rule for a scalar function f(t) times a vector \mathbf{u}(t): \frac{d}{dt} [f(t) \mathbf{u}(t)] = f'(t) \mathbf{u}(t) + f(t) \frac{d\mathbf{u}}{dt}, enabling computation of velocities and accelerations for composite expressions like \mathbf{r}(t) = f(t) \hat{\mathbf{i}} + g(t) \hat{\mathbf{j}} + h(t) \hat{\mathbf{k}}.^[76]^[78]

Dynamics: force, momentum, energy

In Newtonian mechanics, the force \mathbf{F} acting on a body of mass m is defined as the vector that causes the body's acceleration \mathbf{a}, according to Newton's second law: \mathbf{F} = m \mathbf{a}. This equation represents the net force, which is the vector sum \mathbf{F}_{\text{net}} = \sum \mathbf{F}_i of all individual forces \mathbf{F}_i applied to the body. In vector form, Newton's laws describe the dynamics of point particles or rigid bodies under these forces, with the first law stating that if \mathbf{F}_{\text{net}} = \mathbf{0}, then \mathbf{a} = \mathbf{0} and the body moves with constant velocity (inertial motion); the second law as above; and the third law asserting that forces between interacting bodies are equal in magnitude and opposite in direction, \mathbf{F}_{12} = -\mathbf{F}_{21}. These vector formulations enable the analysis of motion in three dimensions by resolving forces into components along orthogonal axes. Linear momentum \mathbf{p} of a body is defined as the vector product \mathbf{p} = m \mathbf{v}, where \mathbf{v} is the velocity vector. The time derivative of momentum gives the net force, \mathbf{F}_{\text{net}} = \frac{d\mathbf{p}}{dt}, which generalizes Newton's second law for variable mass systems, though in standard cases with constant mass, it reduces to m \mathbf{a}. In isolated systems with no external forces (\mathbf{F}_{\text{net}} = \mathbf{0}), the total momentum \sum \mathbf{p}_i is conserved, a direct consequence of Newton's third law ensuring internal forces cancel in pairs. This conservation law applies to collisions and interactions, allowing prediction of post-event velocities from initial conditions. The impulse \mathbf{J} imparted by a variable force over time interval [t_1, t_2] is the integral \mathbf{J} = \int_{t_1}^{t_2} \mathbf{F}(t) \, dt, which equals the change in momentum \Delta \mathbf{p} = \mathbf{p}_f - \mathbf{p}_i, linking instantaneous forces to finite changes in motion. Work W done by a constant force \mathbf{F} on a body displaced by vector \Delta \mathbf{r} is the scalar \mathbf{F} \cdot \Delta \mathbf{r}, representing energy transfer along the displacement direction, with the dot product capturing the force's effective component. For variable forces, work generalizes to the line integral W = \int \mathbf{F} \cdot d\mathbf{r}. The work-energy theorem states that the net work by all forces equals the change in kinetic energy \Delta K, where kinetic energy is K = \frac{1}{2} m \|\mathbf{v}\|^2 = \frac{1}{2} m v^2. This connects dynamics to energy conservation, as in systems where non-conservative forces do work, the mechanical energy changes accordingly, but total momentum conservation holds separately for translational motion./7:_Work_and_Energy/7.4:_Work-Energy_Theorem)

Advanced Topics

Affine versus Euclidean vectors

Euclidean vectors, also known as free vectors, are elements of a vector space that are translation-invariant, meaning their properties remain unchanged under translation. These vectors support standard operations such as addition and scalar multiplication, forming a complete vector space structure without reference to a specific origin. In contrast, affine vectors, often exemplified by position vectors, are associated with specific points in an affine space, where the absence of a distinguished origin prevents direct addition of two position vectors, as such an operation would lack geometric meaning.^[82]^[83] The key distinction lies in the underlying structures: Euclidean vectors operate within a vector space, allowing full linear combinations, while affine vectors inhabit an affine space, which is essentially a vector space "forgotten" its origin, emphasizing parallelism and ratios but not absolute positions. Affine combinations are the appropriate operation here, defined as a linear combination of points \sum_{i=1}^n \alpha_i \mathbf{p}_i where \sum_{i=1}^n \alpha_i = 1; this preserves the barycenter (weighted average position) of the points involved. Such combinations enable expressions like midpoints or weighted averages without shifting the overall location.^[84]^[83] In physics applications, this differentiation is crucial: displacement vectors, representing changes in position (e.g., velocity or force), behave as Euclidean vectors and can be freely added or scaled, reflecting translation invariance. Position vectors, however, are affine, tied to coordinate origins, and their differences yield displacements; adding two positions directly is invalid, as it would imply an arbitrary origin-dependent result. This framework ensures consistent modeling of motion and forces in Euclidean space.^[85]^[86]

Pseudovectors and transformations

In three-dimensional Euclidean space, pseudovectors, also known as axial vectors, are quantities that behave like ordinary vectors under proper rotations but acquire an additional sign change under improper orthogonal transformations, such as reflections or spatial inversions.^[87] This distinction arises because pseudovectors are typically constructed via the cross product of two polar (true) vectors, which introduces a handedness dependent on the orientation of the coordinate system.^[88] Under a general orthogonal transformation represented by a matrix O with \det(O) = \pm 1, a true vector \mathbf{v} transforms as \mathbf{v}' = O \mathbf{v}, while a pseudovector \mathbf{a} transforms as \mathbf{a}' = \det(O) \, O \mathbf{a}.^[89] For proper rotations, where \det(O) = 1, both types transform identically; for improper transformations, where \det(O) = -1, pseudovectors effectively flip sign relative to what a true vector would do.^[90] A canonical example of a pseudovector is angular momentum \mathbf{L}, defined as the cross product \mathbf{L} = \mathbf{r} \times \mathbf{p}, where \mathbf{r} is the position vector and \mathbf{p} is the linear momentum, both polar vectors.^[91] Under spatial inversion (parity transformation, \mathbf{r} \to -\mathbf{r}, \mathbf{p} \to -\mathbf{p}), \mathbf{L} remains unchanged because (-\mathbf{r}) \times (-\mathbf{p}) = \mathbf{r} \times \mathbf{p}, illustrating its even parity compared to the odd parity of polar vectors.^[87] Similarly, torque \boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}, with \mathbf{F} the force (a polar vector), qualifies as a pseudovector and shares this transformation property, reflecting its dependence on the right-hand rule for orientation.^[92] The magnetic field \mathbf{B} provides another physical instance of a pseudovector, evident in the Lorentz force law \mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}), where \mathbf{E} (electric field) and \mathbf{v} (velocity) are polar vectors.^[93] The cross product structure ensures \mathbf{B} transforms with even parity under inversion, remaining invariant while polar vectors reverse, which is crucial for maintaining the form-invariance of Maxwell's equations under orthogonal transformations.^[87] This behavior underscores the role of the determinant in distinguishing pseudovectors, as the sign of \det(O) encodes the parity (orientation-preserving or reversing) of the transformation, ensuring consistent physical interpretations in oriented spaces.^[94]

Generalizations to higher dimensions

Euclidean vectors generalize naturally to n dimensions, where a vector \mathbf{v} belongs to the space \mathbb{R}^n, consisting of ordered n-tuples of real numbers representing components along n orthogonal axes.^[95] This extension preserves the core structure of addition, scalar multiplication, and the Euclidean norm, defined as \|\mathbf{v}\| = \sqrt{\sum_{i=1}^n v_i^2}, which induces a positive-definite metric on the space.^[96] The dot product, or inner product, also generalizes directly as \mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^n u_i v_i, enabling concepts like orthogonality and angles in higher dimensions via the formula \cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}.^[97] In n-dimensional Euclidean space, the vector space \mathbb{R}^n equipped with this inner product forms a real inner product space, where properties such as the Cauchy-Schwarz inequality |\mathbf{u} \cdot \mathbf{v}| \leq \|\mathbf{u}\| \|\mathbf{v}\| hold, ensuring the geometry remains consistent with lower-dimensional cases.^[98] Unlike in three dimensions, there is no natural cross product that yields a vector perpendicular to two given vectors in dimensions greater than 3, as the Hodge dual operator, which defines the cross product in \mathbb{R}^3, does not produce a vector in higher even or odd dimensions beyond that.^[99] Instead, generalizations use the exterior algebra, where the wedge product \mathbf{u} \wedge \mathbf{v} forms a bivector in the second exterior power \bigwedge^2 \mathbb{R}^n, capturing oriented area without reducing to a single vector.^[100] These higher-dimensional Euclidean vectors find applications in data science, where feature vectors in \mathbb{R}^n represent data points with n attributes, facilitating distance-based analyses like clustering via Euclidean distance.^[101] In physics, particularly special relativity, 4-vectors extend the concept to four dimensions but adopt a semi-Euclidean (Minkowski) metric with signature (+,-,-,-), differing from the positive-definite Euclidean metric while still using vector notation for position, momentum, and other quantities.^[102]

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Under rotations, scalars are invariant, vectors transform like coordinates, and tensors transform according to their indices. Space reflection reverses polar ...
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7.4: Angular Momentum - Physics LibreTexts
Mar 4, 2025 · Angular momentum is an example of a mathematical quantity known as an axial vector, or a pseudovector ("regular" vectors like momentum and velocity are ...
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[PDF] FREQUENTLY ASKED QUESTIONS - Duke Physics
Jan 14, 2020 · ... pseudovector”, or “axial vector”. So angular momentum is a pseudovector, whereas linear momentum is known as a “polar vector.” ) Page 2. How ...
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44. Symmetries and the Dirac Monopole - Galileo and Einstein
Vectors that do not change sign under spatial inversion are called pseudovectors or axial vectors. In this context, ordinary vectors are sometimes referred to ...
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[PDF] Classical Electrodynamics - University of Oregon
Apr 9, 2020 · The curl of a vector field transforms as a pseudovector: ^. (∇ × v) ... v is a pseudovector.a. aFor some reason, Belitz writes the ...
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[PDF] Inner-Product Spaces, Euclidean Spaces - CMU Math
Many of the results, for example the Inner-Product In- equality and the Theorem on Subadditivity of Magnitude, remain valid for infinite-dimensional spaces.
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[PDF] Inner Product Spaces and Orthogonality - HKUST Math Department
The vector space Rn with this special inner product (dot product) is called the Euclidean n-space, and the dot product is called the standard inner product on ...
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Inner Product Spaces - Euclidean Tensor Analysis
The simplest, and also the most important, example of an inner product space is the vector space Rn defined earlier, with the inner product ⋅:V×V→R defined as ...
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[PDF] Lecture 5.1: Inner products and Euclidean structure
An inner product on a real vector space X is a symmetric positive-definite bilinear form h−, −i: X × X −→ R. A vector space endowed with an inner product is an ...
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n-dimensional "cross product" reference request - MathOverflow
Apr 17, 2012 · Note: One can express this "cross product" in terms of exterior algebra operations. It is equivalent to ∗(w1∧w ...Generalization of Curl to higher dimensions - MathOverflowWhy is the exterior algebra so ubiquitous? - MathOverflowMore results from mathoverflow.net
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[PDF] 4 Exterior algebra - People
Given this space we can now define our generalization of the cross-product, called the exterior product or wedge product of two vectors. Definition 14 Given ...
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Linear Algebra: Euclidean Vector Space - Towards Data Science
Mar 6, 2023 · Two vectors u = (u₁, u₂) and v = (v₁, v₂) are equal if u₁ = v₁ and u₂ = v₂. · The sum of vector u and v is defined as u + v = (u₁ + v₁, u₂ + v₂).
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17.5: Geometry of Space-time - Physics LibreTexts
Mar 14, 2021 · In 1908 Minkowski reformulated Einstein's Special Theory of Relativity in this 4-dimensional Euclidean space-time vector space and concluded ...