Difference between revisions of "Vector"

Line 2: Line 2:
  
  
 +
== 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''' (and geometric interpretation as the volume of a parallelepiped)
 
  
  
 +
'''Triple Vector Product'''
 
== See Also ==
 
== See Also ==
 
*[[Linear Algebra]]
 
*[[Linear Algebra]]
Line 21: Line 36:
 
== Related threads from AoPS forum ==
 
== Related threads from AoPS forum ==
  
*[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.]
  
  
 
{{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 $\displaystyle n$-dimensional vector can be thought of as an ordered $\displaystyle n$-tuple of numbers within angle brackets. The set of vectors in some space is an example of a vector space.


Description

Every vector $\vec{PQ}$has a starting point $P<x_1, y_1>$ and an endpoint $Q<x_2, y_2>$. 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 $\bold{u}=<u_1,u_2,...,u_n>$ and $\bold{v}=<v_1, v_2,...,v_n>$. The dot product is defined as $\bold{u}\cdot\bold{v}=u_1v_1+u_2v_2+...+u_nv_n$. In two or three dimensions, the dot product has the special geometric property that $\cos{\theta}=\frac{\bold{u}\cdot\bold{v}}{\|\bold{u}\|\|\bold{v}\|}$


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.

Invalid username
Login to AoPS