# Difference between revisions of "Vector"

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− | The word '''vector''' has many different definitions, depending on who is defining it and in what context. Physicists will often refer to a vector as "a quantity with a direction and magnitude." For Euclidean geometers, a vector is essentially a directed line segment. In many situations, a vector is best considered as an n-tuple of numbers (often real or complex). Most generally, but also most abstractly, a vector is any object which is an element of a given vector space | + | The word '''vector''' has many different definitions, depending on who is defining it and in what context. Physicists will often refer to a vector as "a quantity with a direction and magnitude." For Euclidean geometers, a vector is essentially a directed line segment. In many situations, a vector is best considered as an n-tuple of numbers (often real or complex). Most generally, but also most abstractly, a vector is any object which is an element of a given vector space. |

+ | 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>n</math>-dimensional vector can be described in this coordinate form as an ordered <math>n</math>-tuple of numbers within angle brackets or parentheses, <math>(x\,\,y\,\,z\,\,...)</math>. The set of vectors over a [[field]] is called 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. 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 | + | 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. | ||

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

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

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

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#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]]) | #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]]) | ||

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

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

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

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

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'''Dot (Scalar) Product''' | '''Dot (Scalar) Product''' | ||

Consider two vectors <math>\bold{u}=\langle u_1,u_2,\ldots,u_n\rangle</math> and <math>\bold{v}=\langle v_1, v_2,\ldots,v_n\rangle</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+\cdots+u_nv_n</math>. | Consider two vectors <math>\bold{u}=\langle u_1,u_2,\ldots,u_n\rangle</math> and <math>\bold{v}=\langle v_1, v_2,\ldots,v_n\rangle</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+\cdots+u_nv_n</math>. | ||

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

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If <math>\vec{a}=\langle a_1,a_2,a_3\rangle</math> and <math>\vec{b}=\langle b_1,b_2,b_3\rangle</math>, then the cross product of <math>\vec{a}</math> and <math>\vec{b}</math> is given by | If <math>\vec{a}=\langle a_1,a_2,a_3\rangle</math> and <math>\vec{b}=\langle b_1,b_2,b_3\rangle</math>, then the cross product of <math>\vec{a}</math> and <math>\vec{b}</math> is given by | ||

<center><math>\vec{a}\times\vec{b}=\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\end{vmatrix}.</math></center> | <center><math>\vec{a}\times\vec{b}=\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\end{vmatrix}.</math></center> | ||

− | where <math>\hat{i},\hat{j},\hat{k}</math> are [[ | + | where <math>\hat{i},\hat{j},\hat{k}</math> are [[unit vector]]s along the co-ordinate axes. |

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'''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 | '''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> | <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> | ||

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It can also be shown that | It can also be shown that | ||

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*[[Matrix]] | *[[Matrix]] | ||

*[http://www.artofproblemsolving.com/Forum/index.php?f=346\ Matrix-Linear Algebra AOPS forum] | *[http://www.artofproblemsolving.com/Forum/index.php?f=346\ Matrix-Linear Algebra AOPS forum] | ||

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+ | == Discussion == | ||

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

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## Revision as of 20:20, 4 February 2008

The word **vector** has many different definitions, depending on who is defining it and in what context. Physicists will often refer to a vector as "a quantity with a direction and magnitude." For Euclidean geometers, a vector is essentially a directed line segment. In many situations, a vector is best considered as an n-tuple of numbers (often real or complex). Most generally, but also most abstractly, a vector is any object which is an element of a given vector space.

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, .

*An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.*

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.

If and , then the cross product of and is given by

where are unit vectors along the co-ordinate axes.

**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**