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#1 2006-03-30 09:00:45

MathsIsFun
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Vector Formulas

Vector Formulas


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman
 

#2 2006-04-09 02:00:54

ganesh
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Re: Vector Formulas

Let

and
be any two vectors.

Properties:-

(i) Commutativity


(ii) Associativity:-


(iii) Existence of a negative vector:
For every vector
, there is a negative such that


If m and n are scalars,

(iv)


(v)


(vi)


(vii)


(viii)


Character is who you are when no one is looking.
 

#3 2006-04-09 02:20:16

ganesh
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Re: Vector Formulas

Scalar Product


where
is the angle between the two vectors.
If

and

then

and





Character is who you are when no one is looking.
 

#4 2006-04-15 11:05:12

Ricky
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Re: Vector Formulas

Ganesh, you can use \cdot to produce:










Last edited by Ricky (2006-04-15 11:11:54)


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."
 

#5 2006-04-18 01:42:09

ganesh
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Re: Vector Formulas

Cross Product or Vector product
















Character is who you are when no one is looking.
 

#6 2006-04-20 00:26:58

ganesh
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Re: Vector Formulas

Scalar Triple Product

If




then





is also represented as






Interchanging any two vectors in the scalar triple product changes the sign of the scalar triple product.

If any of the two vectors are equal, then



is the volume of the Parellopiped whose coterminus edges are formed by


If
are coplanar,


Character is who you are when no one is looking.
 

#7 2006-04-20 01:45:50

ganesh
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Re: Vector Formulas

Vector Triple Product

The Vector Triple Product of three vectors


is defined as









Vector triple product is not associative.


Character is who you are when no one is looking.
 

#8 2006-04-21 00:49:33

ganesh
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Re: Vector Formulas

Scalar Product of four vectors





Vector product of four vectors

Vector product of four vectors

is given by



Character is who you are when no one is looking.
 

#9 2006-08-06 14:17:40

Zhylliolom
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Re: Vector Formulas

Derivative of a Vector Function

The derivative of a vector function A(u) = A1(u)i + A2(u)j + A3(u)k is given by



Vector derivatives satisfy the following equations(note that Φ is a scalar function):











Integral of a Vector Function

If A(u) = dB/du, then we have the indefinite integral of A(u)



and the definite integral of A(u)

Last edited by Zhylliolom (2006-08-06 16:10:32)

 

#10 2006-08-06 14:22:43

Zhylliolom
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Re: Vector Formulas

The Operator Del/Nabla

The operator del(sometimes called nabla) is defined by



Gradient

The gradient of Φ(x, y, z) is given by



Divergence

The divergence of A is given by



Curl

The curl of A is given by

Last edited by Zhylliolom (2006-08-06 15:17:27)

 

#11 2006-08-06 14:24:02

Zhylliolom
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Re: Vector Formulas

The Laplacian

The Laplacian operator ∇2 may be used on either scalar or vector functions:





The Biharmonic Operator

The biharmonic operator ∇4 on Φ is given by



∇ Formulas

For scalar functions Φ and Θ and vector functions A and B, the following equalities hold:











Last edited by Zhylliolom (2006-08-06 15:16:39)

 

#12 2008-03-12 08:00:51

Daniel123
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Re: Vector Formulas

The equation of a line in vector form

A vector equation of a straight line passing through the point A with with position vector a, and parallel to the vector b, is



A vector equation of a straight line passing through the points C and D, with position vectors c and d respectively, is



Where t is a scalar parameter.

Last edited by Daniel123 (2008-03-12 08:01:33)

 

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