Vectors are usually denoted in
lowercase boldface, as in \mathbf{u}
, \mathbf{v} and \mathbf{w}, or in lowercase italic boldface, as in
a. (
Uppercase letters are typically used to represent
matrices.) Other conventions include \vec{a} or
a, especially in handwriting. Alternatively, some use a
tilde (~) or a wavy underline drawn beneath the symbol, e.g. \underset{^\sim}a, 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 \stackrel{\longrightarrow}{AB} or
AB. In
German literature, it was especially common to represent vectors with small
fraktur letters such as \mathfrak{a}. 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. 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
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 \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 \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: \mathbf{a} = (a_1, a_2, a_3). also written, \mathbf{a} = (a_x, a_y, a_z). This can be generalised to
n-dimensional Euclidean space (or ). \mathbf{a} = (a_1, a_2, a_3, \ldots, a_{n-1}, a_n). These numbers are often arranged into a
column vector or
row vector, particularly when dealing with
matrices, as follows: \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: {\mathbf e}_1 = (1,0,0),\ {\mathbf e}_2 = (0,1,0),\ {\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: \mathbf{a} = (a_1,a_2,a_3) = a_1(1,0,0) + a_2(0,1,0) + a_3(0,0,1), \ or \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, 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 \mathbf{i},\mathbf{j},\mathbf{k} instead (or \mathbf{\hat{x}}, \mathbf{\hat{y}}, \mathbf{\hat{z}}, in which the
hat symbol \mathbf{\hat{}} 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, \mathbf{a} = \mathbf{a}_x + \mathbf{a}_y + \mathbf{a}_z = a_x{\mathbf i} + a_y{\mathbf j} + a_z{\mathbf k}. 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. 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 \mathbf{\hat{x}}, \mathbf{\hat{y}}, \mathbf{\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 (\boldsymbol{\hat{\rho}}, \boldsymbol{\hat{\phi}}, \mathbf{\hat{z}}) or
spherical coordinate system (\mathbf{\hat{r}}, \boldsymbol{\hat{\theta}}, \boldsymbol{\hat{\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
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==