vector, quantity having both magnitude and direction; it may be represented by a directed line segment. Many physical quantities are vectors, e.g., force, velocity, and momentum. Thus, in specifying a force, one must state not only how large it is but also in what direction it acts.

Representation and Reference Systems

The simplest representation of a vector is as an arrow connecting two points. Thus, AB designates the vector represented by an arrow from point A to point B, while BA designates a vector of equal magnitude in the opposite direction, from B to A. In order to compare vectors and to operate on them mathematically, however, it is necessary to have some reference system that determines scale and direction. Cartesian coordinates are often used for this purpose. In the plane, two axes and unit lengths along each axis serve to determine magnitude and direction throughout the plane. For example, if the point A mentioned above has coordinates (2,3) and the point B coordinates (5,7), the size and position of the vector are thus determined. The size of the vector in the x-direction is found by projecting the vector onto the x-axis, i.e., by dropping perpendicular line segments to the x-axis. The length of this projection is simply the difference between the x-coordinates of the two points A and B, or 5 - 2 = 3. This is called the x-component of the vector. Similarly, the y-component of the vector is found to be 7 - 3 = 4. A vector is frequently expressed by giving its components with respect to the coordinate axes; thus, our vector becomes [3,4].

Knowledge of the components of a vector enables one to compute its magnitude—in this case, 5, from the Pythagorean theorem [(3² + 4²)^1/2 = 5)]—and its direction from trigonometry, once the lengths of the sides of the right triangle formed by the vector and its components are known. (Trigonometry can also be used to find the component of the vector as projected in some direction other than the x-axis or y-axis.) Since the vector points from A to B, both its components are positive; if it pointed from B to A, its components would be [-3,-4] but its magnitude and orientation would be the same.

It is obvious that an infinite number of vectors can have the same components [3,4], since there are an infinite number of pairs of points in the plane with x- and y-coordinates whose respective differences are 3 and 4. All these vectors have the same magnitude and direction, being parallel to one another, and are considered equal. Thus, any vector with components a and b can be considered as equal to the vector [a,b] directed from the origin (0,0) to the point (a,b). The concept of a vector can be extended to three or more dimensions.

Addition and Multiplication of Vectors

The addition, or composition, of two vectors can be accomplished either algebraically or graphically. For example, to add the two vectors U [-3,1] and V [5,2], one can add their corresponding components to find the resultant vector R [2,3], or one can graph U and V on a set of coordinate axes and complete the parallelogram formed with U and V as adjacent sides to obtain R as the diagonal from the common vertex of U and V.

Two different kinds of multiplication are defined for vectors in three dimensions. The scalar, or dot, product of two vectors, A and B, is a scalar, or quantity that has a magnitude but no direction, rather than a vector, and is equal to the product of the magnitudes of A and B and the cosine of the angle θ between them, or A · B = ~~pipe~;A~~pipe~; ~~pipe~;B~~pipe~; cos θ. The vector, or cross, product of A and B is a vector, A × B, whose magnitude is equal to ~~pipe~;A~~pipe~; ~~pipe~;B~~pipe~; sin θ and whose orientation is perpendicular to both A and B and pointing in the direction in which a right-hand screw would advance if turned from A to B through the angle θ. The vector product is an example of a kind of multiplication that does not follow the commutative law, since A × B = -B × A.

Vector Analysis and Vector Space

The components of a vector need not be constants but can also be variables and functions of variables. For example, the position of a body moving through space can be described by a vector whose x, y, and z components are each functions of time. The methods of the calculus may be applied to such vector functions, leading to the branch of mathematics known as vector analysis.

The more general extension of vectors leads to the concept of a vector space. A vector space is a set of elements, A, B, C, … , called vectors, for which the operations of addition of vectors and multiplication of a vector by a scalar are defined and which satisfies ten axioms relating to such properties as closure under both operations, associativity, commutativity, and existence of a zero vector, an additive inverse (negative of a vector), and a unit scalar.

In mathematics, a collection of objects called vectors, together with a field of objects (see field theory), known as scalars, that satisfy certain properties. The properties that must be satisfied are: (1) the set of vectors is closed under vector addition; (2) multiplication of a vector by a scalar produces a vector in the set; (3) the associative law holds for vector addition, u + (v + w) = (u + v) + w; (4) the commutative law holds for vector addition, u + v = v + u; (5) there is a 0 vector such that v + 0 = v; (6) every vector has an additive inverse (see inverse function), v + (−v) = 0; (7) the distributive law holds for scalar multiplication over vector addition, math.n(u + v) = math.nu + math.nv; (8) the distributive law also holds for vector multiplication over scalar addition, (math.m + math.n)v = math.mv + math.nv; (9) the associative law holds for scalar multiplication with a vector, (math.mmath.n)v = math.m(math.nv); and (10) there exists a unit vector 1 such that 1v = v. The set of all polynomials in one variable with real coefficients is an example of a vector space.

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In mathematics, a quantity characterized by magnitude and direction. Some physical and geometric quantities, called scalars, can be fully defined by a single number specifying their magnitude in suitable units of measure (e.g., mass in grams, temperature in degrees, time in seconds). Quantities like velocity, force, and displacement must be specified by a magnitude and a direction. These are vectors. A vector quantity can be visualized as an arrow drawn in a specific direction, whose length is equal to the magnitude of the quantity represented. A two-dimensional vector is specified by two coordinates, a three-dimensional vector by three coordinates, and so on. Vector analysis is a branch of mathematics that explores the utility of this type of representation and defines the ways such quantities may be combined. Seealso vector operations.

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In relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations. The usage of the four-vector name tacitly assumes that its components refer to a standard basis. The components transform between these bases as the space and time coordinate differences, $(cDelta\; t,\; Delta\; x,\; Delta\; y,\; Delta\; z)$ under spatial translations, rotations, and boosts (a change by a constant velocity to another inertial reference frame). The set of all such translations, rotations, and boosts (called Poincaré transformations) forms the Poincaré group. The set of rotations and boosts (Lorentz transformations, described by 4×4 matrices) forms the Lorentz group.

This article considers four-vectors in the context of special relativity. Although the concept of four-vectors also extends to general relativity, some of the results stated in this article require modification in general relativity.

Mathematics of four-vectors

A point in Minkowski space is called an "event" and is described in a standard basis by a set of four coordinates such as

$mathbf\{X\}\; :=\; left(X^0,\; X^1,\; X^2,\; X^3\; right)\; =\; left(ct,\; x,\; y,\; z\; right)$

where $mu$ = 0, 1, 2, 3, labels the spacetime dimensions and where c is the speed of light. The definition $X^0\; =\; ct$ ensures that all the coordinates have the same units (of distance). These coordinates are the components of the position four-vector for the event. The displacement four-vector is defined to be an "arrow" linking two events:

$Delta\; mathbf\{X\}:=\; left(Delta\; ct,\; Delta\; x,\; Delta\; y,\; Delta\; z\; right)$

(Note that the position vector is the displacement vector when one of the two events is the origin of the coordinate system. Position vectors are relatively trivial; the general theory of four-vectors is concerned with displacement vectors.)

The (pseudo-) inner product of two four-vectors $mathbf\{U\}$ and $mathbf\{V\}$ is defined (using Einstein notation) as

$$

mathbf{U cdot V} = eta_{mu nu} U^{mu} V^{nu} = left(begin{matrix}U^0 & U^1 & U^2 & U^3 end{matrix} right) left(begin{matrix} 1 & 0 & 0 & 0 0 & -1 & 0 & 0 0 & 0 & -1 & 0 0 & 0 & 0 & -1 end{matrix} right) left(begin{matrix}V^0 V^1 V^2 V^3 end{matrix} right) = U^0 V^0 - U^1 V^1 - U^2 V^2 - U^3 V^3

where η is the Minkowski metric. Sometimes this inner product is called the Minkowski inner product. It is not a true inner product in the mathematical sense because it is not positive definite. Note: some authors define η with the opposite sign:

$eta\_\{mu\; nu\}$

= left(begin{matrix} -1 & 0 & 0 & 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 0 & 0 & 0 & 1 end{matrix} right) in which case

$$

mathbf{U cdot V} = -U^0 V^0 + U^1 V^1 + U^2 V^2 + U^3 V^3

An important property of the inner product is that it is invariant (that is, a scalar): a change of coordinates does not result in a change in value of the inner product.

The inner product is often expressed as the effect of the dual vector of one vector on the other:

$mathbf\{U\; cdot\; V\}\; =\; U^*(mathbf\{V\})\; =\; U\{\_nu\}V^\{nu\}$

Here the $U\_\{nu\}$s are the components of the dual vector $U^*$ of $mathbf\{U\}$ in the dual basis and called the covariant coordinates of $mathbf\{U\}$, while the original $U^nu$ components are called the contravariant coordinates. Lower and upper indices indicate always covariant and contravariant coordinates, respectively.

The relation between the covariant and contravariant coordinates is:

$U\_\{mu\}\; =\; eta\_\{mu\; nu\}\; U^\{nu\}\; ,$.

The four-vectors are arrows on the spacetime diagram or Minkowski diagram. In this article, four-vectors will be referred to simply as vectors.

Four-vectors may be classified as either spacelike, timelike or null. Spacelike, timelike, and null vectors are ones whose inner product with themselves is greater than, less than, and equal to zero respectively.

In special relativity (but not general relativity), the derivative of a four-vector with respect to a scalar (invariant) is itself a four-vector.

Examples of four-vectors in dynamics

When considering physical phenomena, differential equations arise naturally; however, when considering space and time derivatives of functions, it is unclear which reference frame these derivatives are taken with respect to. It is agreed that time derivatives are taken with respect to the proper time (τ). As proper time is an invariant, this guarantees that the proper-time-derivative of any four-vector is itself a four-vector. It is then important to find a relation between this proper-time-derivative and another time derivative (using the time of an inertial reference frame). This relation is provided by the time transformation in the Lorentz transformations and is:

$frac\{d\; tau\}\{dt\}=frac\{1\}\{gamma\}$

where γ is the Lorentz factor. Important four-vectors in relativity theory can now be defined, such as the four-velocity of an $mathbf\{X\}(tau)$ world line is defined by:

$mathbf\{U\}\; :=\; frac\{dmathbf\{X\}\}\{d\; tau\}=\; frac\{dmathbf\{X\}\}\{dt\}frac\{dt\}\{d\; tau\}=\; left(gamma\; c,\; gamma\; mathbf\{u\}\; right)$

where

$u^i\; =\; frac\{dX^i\}\{dt\}$

for i = 1, 2, 3. Notice that

$||\; mathbf\{U\}\; ||^2\; =\; U^\{mu\}\; U\_\{mu\}\; =\; c^2\; ,$

The four-acceleration is given by:

$mathbf\{A\}\; =frac\{dmathbf\{U\}\; \}\{d\; tau\}\; =\; left(gamma\; dot\{gamma\}\; c,\; gamma\; dot\{gamma\}\; mathbf\{u\}\; +\; gamma^2\; mathbf\{dot\{u\}\}\; right)$

Since the magnitude $sqrt\{\; |\; U\_mu\; U^mu\; |\; \}$ of $mathbf\{U\}$ is a constant, the four acceleration is (pseudo-)orthogonal to the four velocity, i.e. the Minkowski inner product of the four-acceleration and the four-velocity is zero:

$A\_mu\; U^mu\; =frac\{1\}\{2\}\; frac\{partial\; (U^mu\; U\_mu)\}\{partial\; tau\; \}\; =\; 0\; ,$

which is true for all world lines.

The four-momentum for a massive particle is given by:

$mathbf\{P\}\; :=m\; mathbf\{U\}\; =\; left(gamma\; mc,\; mathbf\{p\}\; right)$

where m is the invariant mass of the particle and $mathbf\{p\}\; =\; m\; gamma\; mathbf\{u\}$ is the relativistic momentum.

The four-force is defined by:

$mathbf\{F\}\; :=\; frac\; \{d\; mathbf\{P\}\}\; \{d\; tau\}$

For a particle of constant mass, this is equivalent to

$mathbf\{F\}\; =\; m\; mathbf\{A\}\; =\; left(gamma\; dot\{gamma\}\; mc,\; gamma\; mathbf\{f\}\; right)$

where

$mathbf\{f\}\; =\; frac\{dmathbf\{p\}\}\{dt\}\; =\; m\; dot\{gamma\}\; mathbf\{u\}\; +\; m\; gamma\; mathbf\{dot\{u\}\}$.

Physics of four-vectors

The power and elegance of the four-vector formalism may be demonstrated by seeing that known relations between energy and matter are embedded into it.

E = mc²

Here, an expression for the total energy of a particle will be derived. The kinetic energy (K) of a particle is defined analogously to the classical definition, namely as

$frac\{dK\}\{dt\}=\; mathbf\{f\}\; cdot\; mathbf\{u\}$

with f as above. Noticing that $F^mu\; U\_mu\; =\; 0$ and expanding this out we get

$gamma^2\; left(mathbf\{f\}\; cdot\; mathbf\{u\}\; -\; dot\{gamma\}\; mc^2\; right)\; =\; 0$

Hence

$frac\{dK\}\{dt\}\; =\; c^2\; frac\{dgamma\; m\}\{dt\}$

which yields

$K\; =\; gamma\; m\; c^2\; +\; S\; ,$

for some constant S. When the particle is at rest (u = 0), we take its kinetic energy to be zero (K = 0). This gives

$S\; =\; -m\; c^2\; ,$

Thus, we interpret the total energy E of the particle as composed of its kinetic energy K and its rest energy m c². Thus, we have

$E\; =\; gamma\; m\; c^2\; ,$

E² = p²c² + m²c⁴

Using the relation $E\; =\; gamma\; mc^2$, we can write the four-momentum as

$mathbf\{P\}\; =\; left(frac\{E\}\{c\},\; mathbf\{p\}\; right)$.

Taking the inner product of the four-momentum with itself in two different ways, we obtain the relation

$frac\{E^2\}\{c^2\}\; -\; p^2\; =\; P^mu\; P\_mu\; =\; m^2\; U^mu\; U\_mu\; =\; m^2\; c^2$

i.e.

$frac\{E^2\}\{c^2\}\; -\; p^2\; =\; m^2\; c^2$

Hence

$E^2\; =\; p^2\; c^2\; +\; m^2\; c^4\; ,$

This last relation is useful in many areas of physics.

Examples of four-vectors in electromagnetism

Examples of four-vectors in electromagnetism include the four-current defined by

$mathbf\{J\}\; :=\; left(rho\; c,\; mathbf\{j\}\; right)$

formed from the current density j and charge density ρ, and the electromagnetic four-potential defined by

$mathbf\{Psi\}\; :=\; left(phi\; ,\; mathbf\{a\}\; c\; right)$

formed from the vector potential a and the scalar potential $phi\; ,$.

A plane electromagnetic wave can be described by the four-frequency defined as

$mathbf\{N\}\; :=left(nu,\; nu\; mathbf\{n\}\; right)$

where $nu$ is the frequency of the wave and n is a unit vector in the travel direction of the wave. Notice that

$N^mu\; N\_mu\; =\; nu\; ^2\; left(n^2\; -\; 1\; right)\; =\; 0$

so that the four-frequency is always a null vector.

A wave packet of nearly monochromatic light can be characterized by the wave vector, or four-wavevector

$mathbf\{K\}\; =\; left(frac\{omega\}\{c\},\; mathbf\{k\}\; right)\; ,$

See also

• four-velocity
• four-acceleration
• four-momentum
• four-force
• four-current
• electromagnetic four-potential
• four-gradient
• four-frequency
• wave vector
• Basic introduction to the mathematics of curved spacetime
• Minkowski space

References

• Rindler, W. Introduction to Special Relativity (2nd edn.) (1991) Clarendon Press Oxford ISBN 0-19-853952-5