Euclidean vector

class=skin-invert-image|thumb|A vector <math display=inline>\stackrel \rightarrow{a}</math> pointing from point A to point B

In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Euclidean vectors can be added and scaled to form a vector space. A vector quantity is a vector-valued physical quantity, including units of measurement and possibly a support, formulated as a directed line segment. A vector is frequently depicted graphically as an arrow connecting an initial point A with a terminal point B, and denoted by <math display=inline>\stackrel \longrightarrow{AB}.</math>

A vector is what is needed to "carry" the point A to the point B; the Latin word means 'carrier'. It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space.

Vectors play an important role in physics: the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement), their magnitude and direction can still be represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like mathematical objects that describe physical quantities, such as pseudovectors and tensors, transform in a similar way under changes of the coordinate system.

History

The vector concept, as it is known today, is the result of a gradual development over a period of more than 200 years. About a dozen people contributed significantly to its development. In 1835, Giusto Bellavitis abstracted the basic idea when he established the concept of equipollence. Working in a Euclidean plane, he made equipollent any pair of parallel line segments of the same length and orientation. Essentially, he realized an equivalence relation on the pairs of points (bipoints) in the plane, and thus erected the first space of vectors in the plane.

Several other mathematicians developed vector-like systems in the middle of the nineteenth century, including Augustin Cauchy, Hermann Grassmann, August Möbius, Comte de Saint-Venant, and Matthew O'Brien. Grassmann's 1840 work (Theory of the Ebb and Flow) was the first system of spatial analysis that is similar to today's system, and had ideas corresponding to the cross product, scalar product and vector differentiation. Grassmann's work was largely neglected until the 1870s. In pure mathematics, a vector is defined more generally as any element of a vector space. In this context, vectors are abstract entities which may or may not be characterized by a magnitude and a direction. This generalized definition implies that the above-mentioned geometric entities are a special kind of abstract vectors, as they are elements of a special kind of vector space called Euclidean space. This particular article is about vectors strictly defined as arrows in Euclidean space. When it becomes necessary to distinguish these special vectors from vectors as defined in pure mathematics, they are sometimes referred to as geometric, spatial, or Euclidean vectors.

A Euclidean vector may possess a definite initial point and terminal point; such a condition may be emphasized calling the result a bound vector. When only the magnitude and direction of the vector matter, and the particular initial or terminal points are of no importance, the vector is called a free vector. The distinction between bound and free vectors is especially relevant in mechanics, where a force applied to a body has a point of contact (see resultant force and couple).

Two arrows <math>\stackrel {\,\longrightarrow}{AB}</math> and <math>\stackrel {\,\longrightarrow}{A'B'}</math> in space represent the same free vector if they have the same magnitude and direction: that is, they are equipollent if the quadrilateral ABB′A′ is a parallelogram. If the Euclidean space is equipped with a choice of origin, then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin.

The term vector also has generalizations to higher dimensions, and to more formal approaches with much wider applications.

Further information

In classical Euclidean geometry (i.e., synthetic geometry), vectors were introduced (during the 19th century) as equivalence classes under equipollence of ordered pairs of points ; two pairs and being equipollent if the points , in this order, form a parallelogram. Such an equivalence class is called a vector, more precisely, a Euclidean vector. The equivalence class of is often denoted <math>\overrightarrow{AB}.</math>

A Euclidean vector is thus an equivalence class of directed segments with the same magnitude (e.g., the length of the line segment ) and same direction (e.g., the direction from to ). In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific place, in contrast to scalars, which have no direction.

Generalizations

In physics, as well as mathematics, a vector is often identified with a tuple of components, or list of numbers, that act as scalar coefficients for a set of basis vectors. When the basis is transformed, for example by rotation or stretching, then the components of any vector in terms of that basis also transform in an opposite sense. The vector itself has not changed, but the basis has, so the components of the vector must change to compensate. The vector is called covariant or contravariant, depending on how the transformation of the vector's components is related to the transformation of the basis. In general, contravariant vectors are "regular vectors" with units of distance (such as a displacement), or distance times some other unit (such as velocity or acceleration); covariant vectors, on the other hand, have units of one-over-distance such as gradient. If you change units (a special case of a change of basis) from meters to millimeters, a scale factor of 1/1000, a displacement of 1 m becomes 1000 mm—a contravariant change in numerical value. In contrast, a gradient of 1 K/m becomes 0.001 K/mm—a covariant change in value (for more, see covariance and contravariance of vectors). Tensors are another type of quantity that behave in this way; a vector is one type of tensor.

In pure mathematics, a vector is any element of a vector space over some field and is often represented as a coordinate vector. The vectors described in this article are a very special case of this general definition, because they are contravariant with respect to the ambient space. Contravariance captures the physical intuition behind the idea that a vector has "magnitude and direction".

Representations

class=skin-invert-image|right|200px|Vector arrow pointing from A to B

Vectors are usually denoted in lowercase boldface, as in <math>\mathbf{u}</math>, <math>\mathbf{v}</math> and <math>\mathbf{w}</math>, or in lowercase italic boldface, as in a. (Uppercase letters are typically used to represent matrices.) Other conventions include <math>\vec{a}</math> or a, especially in handwriting. Alternatively, some use a tilde (~) or a wavy underline drawn beneath the symbol, e.g. <math>\underset{^\sim}a</math>, which is a convention for indicating boldface type. If the vector represents a directed distance or displacement from a point A to a point B (see figure), it can also be denoted as <math>\stackrel{\longrightarrow}{AB}</math> or AB. In German literature, it was especially common to represent vectors with small fraktur letters such as <math>\mathfrak{a}</math>.

Vectors are usually shown in graphs or other diagrams as arrows (directed line segments), as illustrated in the figure. Here, the point A is called the origin, tail, base, or initial point, and the point B is called the head, tip, endpoint, terminal point or final point. The length of the arrow is proportional to the vector's magnitude, while the direction in which the arrow points indicates the vector's direction.

class=skin-invert-image|right|200px

On a two-dimensional diagram, a vector perpendicular to the plane of the diagram is sometimes desired. These vectors are commonly shown as small circles. A circle with a dot at its centre (Unicode U+2299 ⊙) indicates a vector pointing out of the front of the diagram, toward the viewer. A circle with a cross inscribed in it (Unicode U+2297 ⊗) indicates a vector pointing into and behind the diagram. These can be thought of as viewing the tip of an arrow head on and viewing the flights of an arrow from the back.

class=skin-invert-image|thumb|right|A vector in the Cartesian plane, showing the position of a point A with coordinates (2, 3)

class=skin-invert-image|300px|right

In order to calculate with vectors, the graphical representation may be too cumbersome. Vectors in an n-dimensional Euclidean space can be represented as coordinate vectors in a Cartesian coordinate system. The endpoint of a vector can be identified with an ordered list of n real numbers (n-tuple). These numbers are the coordinates of the endpoint of the vector, with respect to a given Cartesian coordinate system, and are typically called the scalar components (or scalar projections) of the vector on the axes of the coordinate system.

As an example in two dimensions (see figure), the vector from the origin O = (0, 0) to the point A = (2, 3) is simply written as

<math display=block>\mathbf{a} = (2,3).</math>

The notion that the tail of the vector coincides with the origin is implicit and easily understood. Thus, the more explicit notation <math>\overrightarrow{OA}</math> is usually deemed not necessary (and is indeed rarely used).

In three dimensional Euclidean space (or ), vectors are identified with triples of scalar components:

<math display=block>\mathbf{a} = (a_1, a_2, a_3).</math>

also written,

<math display=block>\mathbf{a} = (a_x, a_y, a_z).</math>

This can be generalised to n-dimensional Euclidean space (or ).

<math display="block">\mathbf{a} = (a_1, a_2, a_3, \ldots, a_{n-1}, a_n).</math>

These numbers are often arranged into a column vector or row vector, particularly when dealing with matrices, as follows:

<math display=block>\mathbf{a} =

\begin{bmatrix}

a_1\\

a_2\\

a_3\\

\end{bmatrix} =

[ a_1\ a_2\ a_3 ]^{\operatorname{T.

</math>

Another way to represent a vector in n-dimensions is to introduce the standard basis vectors. For instance, in three dimensions, there are three of them:

<math display=block>{\mathbf e}_1 = (1,0,0),\ {\mathbf e}_2 = (0,1,0),\ {\mathbf e}_3 = (0,0,1).</math>

These have the intuitive interpretation as vectors of unit length pointing up the x-, y-, and z-axis of a Cartesian coordinate system, respectively. In terms of these, any vector a in can be expressed in the form:

<math display=block>\mathbf{a} = (a_1,a_2,a_3) = a_1(1,0,0) + a_2(0,1,0) + a_3(0,0,1), \ </math>

<math display=block>\mathbf{a} = \mathbf{a}_1 + \mathbf{a}_2 + \mathbf{a}_3 = a_1{\mathbf e}_1 + a_2{\mathbf e}_2 + a_3{\mathbf e}_3,</math>

where a1, a2, a3 are called the vector components (or vector projections) of a on the basis vectors or, equivalently, on the corresponding Cartesian axes x, y, and z (see figure), while a1, a2, a3 are the respective scalar components (or scalar projections).

In introductory physics textbooks, the standard basis vectors are often denoted <math>\mathbf{i},\mathbf{j},\mathbf{k}</math> instead (or <math>\mathbf{\hat{x, \mathbf{\hat{y, \mathbf{\hat{z</math>, in which the hat symbol <math>\mathbf{\hat{</math> typically denotes unit vectors). In this case, the scalar and vector components are denoted respectively ax, ay, az, and ax, ay, az (note the difference in boldface). Thus,

<math display=block>\mathbf{a} = \mathbf{a}_x + \mathbf{a}_y + \mathbf{a}_z = a_x{\mathbf i} + a_y{\mathbf j} + a_z{\mathbf k}.</math>

The notation ei is compatible with the index notation and the summation convention commonly used in higher level mathematics, physics, and engineering.

Decomposition or resolution

As explained above, a vector is often described by a set of vector components that add up to form the given vector. Typically, these components are the projections of the vector on a set of mutually perpendicular reference axes (basis vectors). The vector is said to be decomposed or resolved with respect to that set.

class=skin-invert-image|right|thumb|Illustration of tangential and normal components of a vector to a surface.

The decomposition or resolution of a vector into components is not unique, because it depends on the choice of the axes on which the vector is projected.

Moreover, the use of Cartesian unit vectors such as <math>\mathbf{\hat{x, \mathbf{\hat{y, \mathbf{\hat{z</math> as a basis in which to represent a vector is not mandated. Vectors can also be expressed in terms of an arbitrary basis, including the unit vectors of a cylindrical coordinate system (<math>\boldsymbol{\hat{\rho, \boldsymbol{\hat{\phi, \mathbf{\hat{z</math>) or spherical coordinate system (<math>\mathbf{\hat{r, \boldsymbol{\hat{\theta, \boldsymbol{\hat{\phi</math>). The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry, respectively.

The choice of a basis does not affect the properties of a vector or its behaviour under transformations.

A vector can also be broken up with respect to "non-fixed" basis vectors that change their orientation as a function of time or space. For example, a vector in three-dimensional space can be decomposed with respect to two axes, respectively normal, and tangent to a surface (see figure). Moreover, the radial and tangential components of a vector relate to the radius of rotation of an object. The former is parallel to the radius and the latter is orthogonal to it.

In these cases, each of the components may be in turn decomposed with respect to a fixed coordinate system or basis set (e.g., a global coordinate system, or inertial reference frame).

Properties and operations

The following section uses the Cartesian coordinate system with basis vectors

<math display=block>{\mathbf e}_1 = (1,0,0),\ {\mathbf e}_2 = (0,1,0),\ {\mathbf e}_3 = (0,0,1)</math>

and assumes that all vectors have the origin as a common base point. A vector a will be written as

<math display=block>{\mathbf a} = a_1{\mathbf e}_1 + a_2{\mathbf e}_2 + a_3{\mathbf e}_3.</math>

Equality

Two vectors are said to be equal if they have the same magnitude and direction. Equivalently they will be equal if their coordinates are equal. So two vectors

<math display=block>{\mathbf a} = a_1{\mathbf e}_1 + a_2{\mathbf e}_2 + a_3{\mathbf e}_3</math>

and

<math display=block>{\mathbf b} = b_1{\mathbf e}_1 + b_2{\mathbf e}_2 + b_3{\mathbf e}_3</math>

are equal if

Opposite, parallel, and antiparallel vectors

Two vectors are opposite if they have the same magnitude but opposite direction;

Addition and subtraction

The sum of a and b of two vectors may be defined as

<math display="block">\mathbf{a}+\mathbf{b}

=(a_1+b_1)\mathbf{e}_1

+(a_2+b_2)\mathbf{e}_2

+(a_3+b_3)\mathbf{e}_3.</math>

The resulting vector is sometimes called the resultant vector of a and b.

The addition may be represented graphically by placing the tail of the arrow b at the head of the arrow a, and then drawing an arrow from the tail of a to the head of b. The new arrow drawn represents the vector a + b, as illustrated below: The term direction cosine refers to the cosine of the angle between two unit vectors, which is also equal to their dot product.

the determinant is unity, |C| = 1;
the inverse is equal to the transpose;
the rows and columns are orthogonal unit vectors, therefore their dot products are zero.

The advantage of this method is that a direction cosine matrix can usually be obtained independently by using Euler angles or a quaternion to relate the two vector bases, so the basis conversions can be performed directly, without having to work out all the dot products described above.

By applying several matrix multiplications in succession, any vector can be expressed in any basis so long as the set of direction cosines is known relating the successive bases.