Euclidean vector

 ( Kinematics ) 

In Cartesian coordinates, a free vector may be thought of in terms of a corresponding bound vector, in this sense, whose initial point has the coordinates of the origin . It is then determined by the coordinates of that bound vector's terminal point. Thus the free vector represented by (1, 0, 0) is a vector of unit length—pointing along the direction of the positive x-axis.

This coordinate representation of free vectors allows their algebraic features to be expressed in a convenient numerical fashion. For example, the sum of the two (free) vectors (1, 2, 3) and (−2, 0, 4) is the (free) vector $$(1,\; 2,\; 3)\; +\; (-2,\; 0,\; 4)\; =\; (1-2,\; 2+0,\; 3+4)\; =\; (-1,\; 2,\; 7)\backslash ,.$$

In a pseudo-Euclidean space, a vector's squared length can be positive, negative, or zero. An important example is Minkowski space (which is important to our understanding of special relativity).

However, it is not always possible or desirable to define the length of a vector. This more general type of spatial vector is the subject of (for free vectors) and (for bound vectors, as each represented by an ordered pair of "points"). One physical example comes from thermodynamics, where many quantities of interest can be considered vectors in a space with no notion of length or angle. Thermodynamics and Differential Forms

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".

Vectors are usually shown in graphs or other diagrams as arrows (directed ), 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.

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.

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 (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 $$\backslash mathbf\{a\}\; =\; (2,3).$$

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

In three dimensional Euclidean space (or ), vectors are identified with triples of scalar components: $$\backslash mathbf\{a\}\; =\; (a\_1,\; a\_2,\; a\_3).$$ also written, $$\backslash mathbf\{a\}\; =\; (a\_x,\; a\_y,\; a\_z).$$

This can be generalised to n-dimensional Euclidean space (or ). $$\backslash mathbf\{a\}\; =\; (a\_1,\; a\_2,\; a\_3,\; \backslash cdots,\; a\_\{n-1\},\; a\_n).$$

These numbers are often arranged into a column vector or row vector, particularly when dealing with matrices, as follows: $$\backslash mathbf\{a\}\; =$$

 \begin{bmatrix}
   a_1\\
   a_2\\
   a_3\\
 \end{bmatrix} =
 [ a_1\ a_2\ a_3 ]^{\operatorname{T}}.
     

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: $$\{\backslash mathbf\; e\}\_1\; =\; (1,0,0),\backslash \; \{\backslash mathbf\; e\}\_2\; =\; (0,1,0),\backslash \; \{\backslash mathbf\; e\}\_3\; =\; (0,0,1).$$ 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: $$\backslash mathbf\{a\}\; =\; (a\_1,a\_2,a\_3)\; =\; a\_1(1,0,0)\; +\; a\_2(0,1,0)\; +\; a\_3(0,0,1),\; \backslash$$

or $$\backslash mathbf\{a\}\; =\; \backslash mathbf\{a\}\_1\; +\; \backslash mathbf\{a\}\_2\; +\; \backslash mathbf\{a\}\_3\; =\; a\_1\{\backslash mathbf\; e\}\_1\; +\; a\_2\{\backslash mathbf\; e\}\_2\; +\; a\_3\{\backslash mathbf\; e\}\_3,$$

where a₁, a₂, a₃ are called the (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 (or scalar projections).

In introductory physics textbooks, the standard basis vectors are often denoted $\backslash mathbf\{i\},\backslash mathbf\{j\},\backslash mathbf\{k\}$ instead (or $\backslash mathbf\{\backslash hat\{x\}\},\; \backslash mathbf\{\backslash hat\{y\}\},\; \backslash mathbf\{\backslash hat\{z\}\}$, in which the hat symbol $\backslash mathbf\{\backslash hat\{\}\}$ typically denotes ). In this case, the scalar and vector components are denoted respectively a_x, a_y, a_z, and a _x, a _y, a _z (note the difference in boldface). Thus,

$$\backslash mathbf\{a\}\; =\; \backslash mathbf\{a\}\_x\; +\; \backslash mathbf\{a\}\_y\; +\; \backslash mathbf\{a\}\_z\; =\; a\_x\{\backslash mathbf\; i\}\; +\; a\_y\{\backslash mathbf\; j\}\; +\; a\_z\{\backslash mathbf\; k\}.$$

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

The decomposition or resolutionGibbs, J.W. (1901). Vector Analysis: A Text-book for the Use of Students of Mathematics and Physics, Founded upon the Lectures of J. Willard Gibbs, by E.B. Wilson, Chares Scribner's Sons, New York, p. 15: "Any vector coplanar with two non-collinear vectors and may be resolved into two components parallel to and respectively. This resolution may be accomplished by constructing the parallelogram ..." 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 $\backslash mathbf\{\backslash hat\{x\}\},\; \backslash mathbf\{\backslash hat\{y\}\},\; \backslash mathbf\{\backslash hat\{z\}\}$ 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 ($\backslash boldsymbol\{\backslash hat\{\backslash rho\}\},\; \backslash boldsymbol\{\backslash hat\{\backslash phi\}\},\; \backslash mathbf\{\backslash hat\{z\}\}$) or spherical coordinate system ($\backslash mathbf\{\backslash hat\{r\}\},\; \backslash boldsymbol\{\backslash hat\{\backslash theta\}\},\; \backslash boldsymbol\{\backslash hat\{\backslash phi\}\}$). 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 Perpendicular 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).

a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3 \\
     

-(\mathbf{a}\ \mathbf{c}\ \mathbf{b}) = -(\mathbf{b}\ \mathbf{a}\ \mathbf{c}) = -(\mathbf{c}\ \mathbf{b}\ \mathbf{a}).