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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,560

Vector Formulas

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

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 16,455

Let

and be any two vectors.**Properties:-**

(i) Commutativity

(ii) Associativity:-

(iii) Existence of a negative vector:

For every vector

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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**ganesh****Moderator**- Registered: 2005-06-28
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**Scalar Product**

where is the angle between the two vectors.

If

and

then

and

Character is who you are when no one is looking.

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

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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**ganesh****Moderator**- Registered: 2005-06-28
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**Cross Product or Vector product**

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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**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.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 16,455

**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.

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**ganesh****Moderator**- Registered: 2005-06-28
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**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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**Zhylliolom****Real Member**- Registered: 2005-09-05
- Posts: 412

**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) = d**B**/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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**Zhylliolom****Real Member**- Registered: 2005-09-05
- Posts: 412

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

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

**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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