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.
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.
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.
See P. Gustyatnikov and S. Reznichenko, Vector Algebra (1988); J. E. Marsden and A. Tromba, Vector Calculus (1988).
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.
where = 0, 1, 2, 3, labels the spacetime dimensions and where c is the speed of light. The definition 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:
(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.)
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:
The inner product is often expressed as the effect of the dual vector of one vector on the other:
Here the s are the components of the dual vector of in the dual basis and called the covariant coordinates of , while the original 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:
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.
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:
for i = 1, 2, 3. Notice that
The four-acceleration is given by:
Since the magnitude of 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:
which is true for all world lines.
The four-momentum for a massive particle is given by:
The four-force is defined by:
For a particle of constant mass, this is equivalent to
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.
with f as above. Noticing that and expanding this out we get
for some constant S. When the particle is at rest (u = 0), we take its kinetic energy to be zero (K = 0). This gives
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
Using the relation , we can write the four-momentum as
Taking the inner product of the four-momentum with itself in two different ways, we obtain the relation
This last relation is useful in many areas of physics.
Examples of four-vectors in electromagnetism include the four-current defined by
formed from the current density j and charge density ρ, and the electromagnetic four-potential defined by
formed from the vector potential a and the scalar potential .
A plane electromagnetic wave can be described by the four-frequency defined as
where is the frequency of the wave and n is a unit vector in the travel direction of the wave. Notice that
so that the four-frequency is always a null vector.