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Vector

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Published on Thursday, May 15, 1997 by Gideon Ariel

Vector

In physics and engineering, the word vector typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a "magnitude" and a "direction".

A common example of a vector is force � it has a magnitude and an orientation in three dimensions (or however many spatial dimensions one has), and multiple forces sum according to the parallelogram law.

Definitions

Informally, a vector is a quantity characterized by a number (indicating magnitude) and a direction, often represented graphically by an arrow. Examples are "moving north at 90 km/h" or "pulling towards the center of Earth with a force of 70 newtons".

Examples of vectors include displacement, velocity, electric field, momentum, force, and acceleration.

Vectors can be contrasted with scalar quantities such as distance, speed, energy, time, temperature, charge, power, work, and mass, which have magnitude, but no direction (they are invariant under coordinate rotations). The magnitude of any vector is a scalar.

Representation of a vector

Symbols standing for vectors are usually printed in boldface as a. The length or magnitude or norm of the vector a is denoted by |a|.

Vectors are usually shown in graphs or other diagrams as arrows, as illustrated below:

Here the point A is called the tail, base, start, or origin; point B is called the head, tip, endpoint, or destination. The length of the arrow represents the vector's magnitude, while the direction in which the arrow points represents the vector's direction.

In order to calculate with vectors, the graphical representation is too cumbersome. Vectors in a n-dimensional Euclidean space can be represented as a linear combination of n mutually perpendicular unit vectors. Let's consider R3 as an example. In R3, we usually denote the unit vectors parallel to the x-, y- and z-axes by i, j and k respectively. Any vector a in R3 can be written as a = a1i + a2j + a3k with real numbers a1, a2 and a3 which are uniquely determined by a. Sometimes a is then also written as a 3-by-1 or 1-by-3 matrix:

even though this notation suppresses the dependence of the coordinates a1, a2 and a3 on the specific choice of coordinate system i, j and k.

See also

2DTwo-dimensional coordinate system
3DThree-dimensional coordinate system
AngleDefinition of an angle
AxisDefinition of Cartesian axis
Cartesian geometryWhat is Cartesian geometry?
Coordinate systemDefinition of coordinates
CurveDefinition of a curve
DistanceDefinition of distance
Euclidean geometryWhat is Euclidean geometry?
GeometryDefinition of geometry
LengthDefinition of length
LineDefinition of a line
OriginDefinition of origin in a Cartesian coordinate system
Perspective projectionDefinition of perspective projection
Planar homographyDefinition of planar homography
PlaneDefinition of a plane
PointDefinition of a point
Point (kinematics)Definition of a point (kinematics)
Projective geometryWhat is projective geometry?
Segment (kinematics)Definition of a segment (kinematics)
Vanishing pointsDefinition of vanishing points and vanishing lines in perspective projection
VectorDefinition of a vector

Reference: /wizard/manual/concepts/geometry.vector.html
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