Difference between revisions of "Vector"
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== Vector Operations == | == Vector Operations == | ||
| − | '''Dot (Scalar) Product''' (proof as well?) | + | '''Dot (Scalar) Product''' (proof as well? ) (--use law of cosines; I'm not good at proofs--from Aryth) |
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>. | 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>. | ||
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| − | '''Triple Vector Product''' | + | '''Triple Vector Product''' |
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== See Also == | == See Also == | ||
*[[Linear Algebra]] | *[[Linear Algebra]] | ||
Revision as of 16:08, 11 August 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? ) (--use law of cosines; I'm not good at proofs--from Aryth)
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
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