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 17:46, 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
[hide]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
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