# Difference between revisions of "Vector"

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+ | == Description == | ||

+ | Every vector <math>\vec{PQ}</math>has a starting point <math>P<x_1, y_1></math> and an endpoint <math>Q<x_2, y_2></math>. 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 == | == Properties of Vectors == | ||

+ | (i) | ||

+ | (ii) | ||

+ | |||

+ | (iii) | ||

+ | |||

+ | (iv) | ||

+ | |||

+ | ... | ||

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

'''Dot (Scalar) Product''' (proof as well?) | '''Dot (Scalar) Product''' (proof as well?) | ||

+ | |||

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

+ | In two or three dimensions, the dot product has the special geometric property that <math>\cos{\theta}=\frac{\bold{u}\cdot\bold{v}}{\|\bold{u}\|\|\bold{v}\|}</math> | ||

+ | |||

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

+ | |||

'''Triple Scalar product''' | '''Triple Scalar product''' | ||

− | |||

− | |||

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

== See Also == | == See Also == | ||

*[[Linear Algebra]] | *[[Linear Algebra]] | ||

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== Related threads from AoPS forum == | == Related threads from AoPS forum == | ||

− | *[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=89911 | + | *[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=89911\ This is a thread about what vectors are.] |

{{stub}} | {{stub}} |

## Revision as of 17:46, 4 July 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

(i)

(ii)

(iii)

(iv)

...

## Vector Operations

**Dot (Scalar) Product** (proof as well?)

Consider two vectors and . The dot product is defined as . In two or three dimensions, the dot product has the special geometric property that

**Cross (Vector) Product**

**Triple Scalar product**

**Triple Vector Product**

## See Also

## Related threads from AoPS forum

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