# Difference between revisions of "Vector"

(→Vector Operations) |
(more work is needed) |
||

Line 1: | Line 1: | ||

− | A '''vector''' is a [[magnitude]] with a [[direction]]. | + | 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 <math>(x,y)</math>. An <math>\displaystyle n</math>-dimensional vector can be described in this coordinate form as an ordered <math>\displaystyle n</math>-tuple of numbers within angle brackets or parentheses, <math>(x\,\,y\,\,z\,\,...)</math>. The set of vectors in some space is an example of a [[vector space]]. |

== Description == | == Description == | ||

− | Every vector <math>\vec{PQ}</math>has a starting point <math>P\langle x_1, y_1\rangle</math> and an endpoint <math>Q\langle x_2, y_2\rangle</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. | + | Every vector <math>\vec{PQ}</math>has a starting point <math>P\langle x_1, y_1\rangle</math> and an endpoint <math>Q\langle x_2, y_2\rangle</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. This is why we only require <math>n</math> values for an <math>n</math> dimensional vector written in the form <math>(x\,\,y\,\,z\,\,...)</math>. The magnitude of a vector, denoted <math>||\vec{v}||</math>, is found simply by |

using the distance formula. | using the distance formula. | ||

+ | |||

+ | == Addition of Vectors == | ||

+ | For vectors <math>\vec{v}</math> and <math>\vec{w}</math>, with angle <math>\theta</math> formed by them, <math>(\vec{v}+\vec{w})^2=||\vec{v}||^2+||\vec{w}||^2-2||\vec{v}||||\vec{w}||\cos\theta</math>. | ||

+ | ***pictures would be helpful here*** | ||

+ | |||

+ | Form this it is simple to derive that for a real number <math>c</math>, <math>c\vec{v}</math> is the vector <math>\vec{v}</math> with magnitude multiplied by <math>c</math>. Negative <math>c</math> corresponds to opposite directions. | ||

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

+ | For vectors <math>\vec{v}</math> and <math>\vec{w}</math>, | ||

+ | |||

(i) | (i) | ||

## Revision as of 11:33, 1 October 2006

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 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. 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***

Form 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

For vectors and ,

(i)

(ii)

(iii)

(iv)

...

## 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

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