# Difference between revisions of "Vector"

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== Properties of Vectors == | == Properties of Vectors == | ||

− | Since a [[vector space]] is defined over a [[field]] <math>K</math>, it is logically inherent that vectors have the same properties as field | + | Since a [[vector space]] is defined over a [[field]] <math>K</math>, it is logically inherent that vectors have the same properties as those elements in a field. |

− | For vectors <math>\vec{ | + | For any vectors <math>\vec{x}</math>, <math>\vec{y}</math>, <math>\vec{z}</math>, and real numbers <math>a,b</math>, |

− | ( | + | #<math>\vec{x}+\vec{y}=\vec{y}+\vec{x}</math> ([[Commutative]] in +) |

− | ( | + | #<math>(\vec{x}+\vec{y})+\vec{z}=\vec{x}+(\vec{y}+\vec{z})</math> ([[Associative]] in +) |

− | ( | + | #There exists the zero vector <math>\vec{0}</math> such that <math>\vec{x}+\vec{0}=\vec{x}</math> ([[Additive identity]]) |

− | ( | + | #For each <math>\vec{x}</math>, there is a vector <math>\vec{y}</math> such that <math>\vec{x}+\vec{y}=\vec{0}</math> ([[Additive inverse]]) |

− | + | #<math>1\vec{x}=\vec{x}</math> (Unit scalar identity) | |

+ | |||

+ | #<math>(ab)\vec{x}=a(b\vec{x})</math> ([[Associative]] in scalar) | ||

+ | |||

+ | #<math>a(\vec{x}+\vec{y})=a\vec{x}+a\vec{y}</math> ([[Distributive]] on vectors) | ||

+ | |||

+ | #<math>(a+b)\vec{x}=a\vec{x}+b\vec{x}</math> ([[Distributive]] on scalars) | ||

== Vector Operations == | == Vector Operations == |

## Revision as of 13:07, 16 November 2007

A **vector** is a magnitude with a direction. A vector is usually graphically represented as an arrow. Vectors can be uniquely described in many ways. The two most common is (for 2-dimensional vectors) by describing it with its length (or magnitude) and the angle it makes with some fixed line (usually the x-axis) or by describing it as an arrow beginning at the origin and ending at the pint . An -dimensional vector can be described in this coordinate form as an ordered -tuple of numbers within angle brackets or parentheses, . The set of vectors over a field is called a vector space.

## Contents

## Description

Every vector has a starting point and an endpoint . Since the only thing that distinguishes one vector from another is its magnitude,i.e. length, and direction, vectors can be freely translated about a plane without changing them. Hence, it is convenient to consider a vector as originating from the origin. This way, two vectors can be compared only by looking at their endpoints. This is why we only require values for an dimensional vector written in the form . The magnitude of a vector, denoted , is found simply by using the distance formula.

## Addition of Vectors

For vectors and , with angle formed by them, .

- pictures would be helpful here***

From this it is simple to derive that for a real number , is the vector with magnitude multiplied by . Negative corresponds to opposite directions.

## Properties of Vectors

Since a vector space is defined over a field , it is logically inherent that vectors have the same properties as those elements in a field.

For any vectors , , , and real numbers ,

- (Commutative in +)

- (Associative in +)

- There exists the zero vector such that (Additive identity)

- For each , there is a vector such that (Additive inverse)

- (Unit scalar identity)

- (Associative in scalar)

- (Distributive on vectors)

- (Distributive on scalars)

## Vector Operations

**Dot (Scalar) Product**
Consider two vectors and in . The dot product is defined as .

**Cross (Vector) Product**
The cross product between two vectors and in is defined as the vector whose length is equal to the area of the parallelogram spanned by and and whose direction is in accordance with the right-hand rule.

**Triple Scalar product** The triple scalar product of three vectors is defined as . Geometrically, the triple scalar product gives the signed area of the parallelpiped determined by and . It follows that

It can also be shown that

**Triple Vector Product**

## See Also

## Related threads from AoPS forum

*This article is a stub. Help us out by expanding it.*