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 18:46, 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.
