Page 267 - Well Logging and Formation Evaluation

Additional Mathematics Theory 257

In particular, it is worth noting that 68.3% of values will fall within 1
standard deviation (SD is the square root of the variance), 95.4% within
2 SD and 99.7% within 3 SD of the mean.

A4.4 VECTOR MECHANICS

An understanding of the basics of vector mechanics is essential to being
able to deal with components in the highside refence system, as discussed
in Chapter 12. Consider a cartesian reference system defined by three
orthogonal axes: x, y, and z. A vector is essentially just a way of writing
directions, in terms of how far you have to travel in the x, y, and z direc-
tions to get from one point to another in space. Hence the vector linking
the origin to the point A located at (a 1, a 2 , a 3 ) is denoted by a, and has
the components:

Ê a1 ˆ
Á a2˜
Á ˜
Ë a3¯

2
2
The length of the vector, denoted by a, is given by ( a 1 + a 2 + a 3 ) . A
2
unit vector is one that has a length of 1. To convert a to a unit vector, each
2
2
2
component would have to be divided by ( a 1 + a 2 + a 3 ) and the vector
would then be designated by â. Taking the scalar product (sometimes
called the dot product) of two vectors allows the angle between them to
be determined:
ac = ac q 1 * + a 2 *c 2 + a 3 * ) (A4.3)
.
* *cos( ) = (a c 1
c 3
where a, c are the magnitude of a and c, and q is the angle between the
vectors. The vector product of two vectors (sometimes called the cross
product) generates a new vector that is orthogonal (i.e., at right angles)
to both. Hence:

Ê a1 ˆ Ê c1 ˆ Ê a2* c3 - a3* c2 ˆ
Á a2˜ Ÿ Á c2˜ = Á a3* c1- a1* c3 ˜ (A4.4)
Á ˜ Á ˜ Á ˜
Ë a3¯ Ë c3¯ Ë a1* c2 - a2* c1 ¯

The vector defined by a Ÿ c has magnitude a*c*sin(q). The direction
of the vector product will follow a right-hand corkscrew rule as one goes