Euclidean Vector - Representations

Representations

Vectors are usually denoted in lowercase boldface, as a or lowercase italic boldface, as a. (Uppercase letters are typically used to represent matrices.) Other conventions include or a, especially in handwriting. Alternatively, some use a tilde (~) or a wavy underline drawn beneath the symbol, 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 or AB. Especially in literature in German it was common to represent vectors with small fraktur letters as .

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; 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, sometimes a vector perpendicular to the plane of the diagram is 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 vanes 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

The notion that the tail of the vector coincides with the origin is implicit and easily understood. Thus, the more explicit notation is usually not deemed necessary and very rarely used.

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

also written

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

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:

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:

or

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 instead denoted (or, in which the hat symbol ^ typically denotes unit vectors). 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,

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