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#1 2006-03-29 10:00:45

MathsIsFun
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Registered: 2005-01-21
Posts: 7,535

Vector Formulas

Vector Formulas


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

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#2 2006-04-08 04:00:54

ganesh
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Registered: 2005-06-28
Posts: 13,595

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.

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#3 2006-04-08 04:20:16

ganesh
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Registered: 2005-06-28
Posts: 13,595

Re: Vector Formulas

Scalar Product


where
is the angle between the two vectors.
If

and

then

and


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#4 2006-04-14 13:05:12

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: Vector Formulas

Ganesh, you can use \cdot to produce:


Last edited by Ricky (2006-04-14 13:11:54)


"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..."

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#5 2006-04-17 03:42:09

ganesh
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Registered: 2005-06-28
Posts: 13,595

Re: Vector Formulas

Cross Product or Vector product






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#6 2006-04-19 02:26:58

ganesh
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Registered: 2005-06-28
Posts: 13,595

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,


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#7 2006-04-19 03:45:50

ganesh
Moderator
Registered: 2005-06-28
Posts: 13,595

Re: Vector Formulas

Vector Triple Product

The Vector Triple Product of three vectors


is defined as


Vector triple product is not associative.


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#8 2006-04-20 02:49:33

ganesh
Moderator
Registered: 2005-06-28
Posts: 13,595

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.

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#9 2006-08-05 16:17:40

Zhylliolom
Real Member
Registered: 2005-09-05
Posts: 412

Re: Vector Formulas

Derivative of a Vector Function

The derivative of a vector function A(u) = A[sub]1[/sub](u)i + A[sub]2[/sub](u)j + A[sub]3[/sub](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 (2006-08-05 18:10:32)

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#10 2006-08-05 16:22:43

Zhylliolom
Real Member
Registered: 2005-09-05
Posts: 412

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 (2006-08-05 17:17:27)

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#11 2006-08-05 16:24:02

Zhylliolom
Real Member
Registered: 2005-09-05
Posts: 412

Re: Vector Formulas

The Laplacian

The Laplacian operator ∇[sup]2[/sup] may be used on either scalar or vector functions:

The Biharmonic Operator

The biharmonic operator ∇[sup]4[/sup] 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 (2006-08-05 17:16:39)

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#12 2008-03-11 09:00:51

Daniel123
Member
Registered: 2007-05-23
Posts: 663

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 (2008-03-11 09:01:33)

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