Mathematical Physics Vol 1

Chapter 1. Vector algebra

14

n-dimensional space. Thus, in a real one-dimensional space (which is geometrically represented by a straight line), to each point a real number is assigned, whose absolute value is the distance (we will define the general term “distance” later) from a predetermined point, for example O , called the origin of the coordinate system. In addition to the origin, it is necessary to determine the unit of distance (the distance to which all other distances shall be compared). To that end, a point A is selected, and the distance OA is considered to be the unit distance. Let P be an arbitrary point, then the number x , assigned to the point P , is defined as follows

OP OA

| x | =

(1.1)

.

If the point is to the right of point O (in our example the point P inFigure 1.1), a plus sign (+) is assumed, namely x > 0, and if it is to the left (in our example the point Q in Figure 1.1) than the sign (–) is assumed, namely x < 0.

Q

A

P

x

O

−

+

Figure 1.1: Oriented straight line.

In this way we determine the direction of the "movement" of a point, and an oriented straight line called the axis is obtained. This orientation is denoted by an arrow indicating the direction in which the numbers are growing. In the real two-dimensional space an ordered pair of real numbers is assigned to each point, with respect to two corresponding lines X 1 and X 2 that intersect at point O (Fig. 1.2). This point is called the origin of the coordinate system .

X 2

X 2

M ′

2

M 2

P

P

B

O

O

X 1

M ′

X 1

M 1

A

1

(a)

(b)

Figure 1.2: Two ways for determining the position of a point.

In this it is also necessary to define a unit of distance, for each axis separately, which means that these distances do not necessarily have to be the same. The pair of these axes, with units of distance OA and OB , represents axes of the coordinate system in the plane. To each point P in the plane an ordered par of real numbers ( x 1 , x 2 ) is assigned, which are called the coordinates of that point , and which are determined as follows. The straight line, which passes through point P , and is parallel to the X 2 –axis, intersects the X 1 –axis at point M 1 , while the straight line parallel to the X 1 –axis, intersects the X 2 –axis at point M 2 (Fig. 1.2(a)).