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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)
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#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
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#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
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#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)