Page 483 - House of Maxell

APPENDIX

463

Introduction
For certainly, the set of Cartesian unit vectors { , , } in the following text is assumed to
0
0
0
be perpendicular and right-handed as Figure A1 demonstrates

Figure A1 Right-handed coordinate system illustration
Vectors Definitions

Vectors are directed line segments in two- or three-dimensional space used to represent
distance, velocity, forces, etc. They have a point of attachment, i.e. in Figure A1, a direction,
0
and a magnitude. Sometimes we will use the short matrix notation of radius or position vector

(, , ) = (, , ) + (, , ) + (, , ) (A.1)
0 0 0
as = � , , �. The magnitude of the vector is denoted || = � + + . Here the set
2
2
2

{ , , } consists of three basis mutually orthogonal unit vectors | | = | | = | | = 1
0
0
0
0
0
0
and ∘ = ∘ = ∘ = 0. The direction of the vector is the unit vector =
0 0 0 0 0 0 0
|| and | | = 1. For simplicity, it was assumed that = (0,0,0) is the origin of the
⁄
0
0
Cartesian coordinate system. In two-dimensional space, the vectors and points of attachment
have one of projections is equal zero.
Parametric Curve
Evidently, the location of a point in space can be defined by the unique position
vector (, , (, )). If so, the tip of this vector can trace the point moving along a curve =
(, ) as it is illustrated in Figure A2a. The plotted in green curve is bounded by the initial
point a and end point b. It is oriented meaning that a consistent direction (like a clockwise green
arrow) is defined along the curve from the initial point to its end point. The curve of non-zero
length with coinciding initial and end point (i.e., a curve starts and ends at the same point)
generally is called closed.
Unluckily, the class of curves that can be represented analytically as a function of and are
quite restricted. We can get more freedom presenting the parametric representation of the curve
defining the single variable t such way that