Explanation:
Euclidean vector
Illustration
of a vector
A
vector going from A to B
In elementary mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or - as here - simply a
vector) is a geometric object that has both a magnitude (or length) and direction. A
Euclidean vector is frequently represented by a line segment with a definite direction, or
graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by
Overview
A vector is a geometric entity
characterized by a magnitude (in mathematics a number, in physics a number times a unit) and a
direction. In rigorous mathematical treatments, a vector is defined as a
directed line segment, or arrow, in a Euclidean space. When it becomes
necessary to distinguish it from vectors as defined elsewhere,
this is sometimes referred to as a geometric, spatial, or Euclidean vector.
As an arrow in Euclidean space, a vector possesses a definite initial point and terminal
point. Such a vector is called a bound
vector. When only the magnitude and direction of the vector matter, then
the particular initial point is of no importance, and the vector is called a free vector. Thus two arrows and in
space represent the same free vector if they have the same magnitude and
direction: that is, they are equivalent 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.
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.
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 indicate the vector's direction.
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
A
vector in the Cartesian plane, showing the position of a point A with
coordinates (2, 3).
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:
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 a_{x}, a_{y}, a_{z}, and a_{x}, a_{y}, a_{z}.
Thus,
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
and
are equal if
(Our solved example in mathguru.com uses
this concept)
http://en.wikipedia.org/wiki/Euclidean_vector
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