# 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 ï¿½ 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 **R**^{3} as an example. In **R**^{3},
we usually denote the unit vectors parallel to the *x*-, *y*- and *z*-axes
by **i**, **j** and **k** respectively. Any vector **a** in **R**3
can be written as **a** = *a*_{1}**i**
+ *a*_{2}**j** + *a*_{3}**k**
with real numbers *a*_{1}, *a*_{2}
and *a*_{3} 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 *a*_{1},
*a*_{2} and *a*_{3}
on the specific choice of coordinate system **i**, **j** and **k**.

**See also**

2D | Two-dimensional coordinate system |

3D | Three-dimensional coordinate system |

Angle | Definition of an angle |

Axis | Definition of Cartesian axis |

Cartesian geometry | What is Cartesian geometry? |

Coordinate system | Definition of coordinates |

Curve | Definition of a curve |

Distance | Definition of distance |

Euclidean geometry | What is Euclidean geometry? |

Geometry | Definition of geometry |

Length | Definition of length |

Line | Definition of a line |

Origin | Definition of origin in a Cartesian coordinate system |

Perspective projection | Definition of perspective projection |

Planar homography | Definition of planar homography |

Plane | Definition of a plane |

Point | Definition of a point |

Point (kinematics) | Definition of a point (kinematics) |

Projective geometry | What is projective geometry? |

Segment (kinematics) | Definition of a segment (kinematics) |

Vanishing points | Definition of vanishing points and vanishing lines in perspective projection |

Vector | Definition of a vector |