Math Is Fun Forum

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Vector Formulas

Vector Formulas

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

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,422

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)

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,422

Re: Vector Formulas

Scalar Product

where

is the angle between the two vectors.
If

and

then

and

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Ricky

Moderator

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)

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"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..."

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,422

Re: Vector Formulas

Cross Product or Vector product

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,422

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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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,422

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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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,422

Re: Vector Formulas

Scalar Product of four vectors

Vector product of four vectors

Vector product of four vectors

is given by

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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)

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)

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)

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)

Monox D. I-Fly

Member

From: Indonesia

Registered: 2015-12-02

Posts: 2,000

Re: Vector Formulas

Where is the supposed "c" in the Left Hand Side?

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