# Difference between revisions of "Vector"

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

− | '''Dot (Scalar) Product''' | + | '''Dot (Scalar) Product''' |

+ | Consider two vectors <math>\bold{u}=<u_1,u_2,\ldots,u_n></math> and <math>\bold{v}=<v_1, v_2,\ldots,v_n></math> in <math>\mathbb{R}^n</math>. The dot product is defined as <math>\bold{u}\cdot\bold{v}=u_1v_1+u_2v_2+...+u_nv_n</math>. | ||

− | |||

− | |||

+ | '''Cross (Vector) Product''' | ||

+ | The cross product between two vectors <math>\bold{a}</math> and <math>\bold{b}</math> in <math>\mathbb{R}^3</math> is defined as the vector whose length is equal to the area of the parallelogram spanned by <math>\bold{a}</math> and <math>\bold{b}</math> and whose direction in accordance with the [[right-hand rule]]. | ||

− | ''' | + | '''Triple Scalar product''' The triple scalar product of three vectors <math>\bold{a,b,c}</math> is defined as <math>(\bold{a}\times\bold{b})\cdot \bold{c}</math>. Geometrically, the triple scalar product gives the signed area of the parallelpiped determined by <math>\bold{a,b}</math> and <math>\bold{c}</math>. It follows that |

+ | |||

+ | <center><math>(\bold{a}\times\bold{b})\cdot \bold{c} = (\bold{c}\times\bold{a})\cdot \bold{b} = (\bold{b}\times\bold{c})\cdot \bold{a}.</math></center> | ||

− | + | It can also be shown that | |

+ | <center><math>(\bold{a}\times\bold{b})\cdot \bold{c} = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}.</math></center> | ||

'''Triple Vector Product''' | '''Triple Vector Product''' |

## Revision as of 22:07, 30 September 2006

A **vector** is a magnitude with a direction. Much of physics deals with vectors. An -dimensional vector can be thought of as an ordered -tuple of numbers within angle brackets. The set of vectors in some space is an example of 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. The magnitude of a vector, denoted is found simply by using the distance formula.

## Properties of Vectors

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## 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 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

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