Chapter One
1.2. Mathematical Foundations : An Introduction to Vector Product
As you all know, a vector is a directed quantity, e.g. velocity, acceleration, momentum or force. In the mathematical formalism (in vector calculus), a vector can be described by a character with an arrow on its top; we used vectors in the equation of the Lorentz-force, Sec.1.1 (Lorentz-Force). Like numbers, every two vectors can be added together. But there are a few more operations, e.g. scalar product or vector product ... here we will deal with the vector product.
The Lorentz-force is one of the simplest examples for a vector product in nature:
The vector product of two vectors 
, 
can be expressed
as: 
= 
,
where the quantity of is defined by:
| | = |
| : = | | | |
. | sin(
(,)) | .
Vector is orientated perpendicular to
and . Moreover, the vectors
, and describe
a right-hand system [Froehner (1998)]. Such
a system is similar to the cartesian system of coordinates. For instance, the vector product of the unit vectors:
1 2
= 3 ; 3 1
= 2 ; 2 3
= 1 .
Per definition, the vector product is anticommutative:
=
- .
Try to
proof the following rules on your own:
= 
; if 
, then
= a b ;
(
)
= ( )
= ( ) ;
( )
(
) ;
(
+ ) =
+ .
How can we calculate the vector product
of two vectors if they can be written as linear combinations of orthonormal vectors? (A system of orthonormal vectors is a set of mutually perpendicular
vectors 1 ,2 ,3 of
unit length)
In short: = a11 +
a22 + a33 ,
= b11 +
b22 + b33 .
Then we
have: = (a11
+ a22 + a33)
(b11
+ b22 + b33)
=
a1b1(1 1) +
a1b2(1 2) +
a1b3(1 3)
+
a2b1(2 1) +
a2b2(2 2) +
a2b3(2 3)
+
a3b1(3 1) +
a3b2(3 2) +
a3b3(3 3)
=
a1b23 - a1b32 -
a2b13 + a2b31 +
a3b12 - a3b21 .
And therefore:
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