Mathematical Physics Vol 1

1.3 Vector algebra

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Coordinates x 1 i x 2 are defined by:

OM 1 OA

OM 2 OB

| x 1 | =

, | x 2 | =

,

where the sign for x 1 and x 2 is determined in the same way as in the one-dimensional space. By this procedure, an ordered pair of numbers ( x 1 , x 2 ) can uniquely be assigned to each point P from the plane (with respect to the given coordinate axes), thus defining the coordinate system of two-dimensional space. This procedure can be generalized and applied to the n –dimensional space ( n > 2). If the angle between the straight axes is 90 ◦ , then such a coordinate system is called Cartesian coordinate system or rectangular (orthogonal) coordinate system. R Note that the procedure for assigning an ordered pair of numbers to a point described above is not the only one used. Namely, it is also possible to draw straight lines from point P that are perpendicular to the corresponding axes (Fig. 1.2(b)), thus obtaining points M ′ 1 i M ′ 2 . In that case the point P has coordinates x ′ 1 and x ′ 2 , defined by: In the special case of Cartesian coordinate system the pairs of numbers ( x 1 , x 2 ) and ( x ′ 1 , x ′ 2 ) are the same. In addition to these procedures for assigning coordinates other procedures are also possible, but these two are generally used in practice. In the previous definitions, the term distance was used, which has so far not been defined. It should be noted that, depending on the expression that defines the distance between two points, different spaces (in mathematical terms) can be distinguished. Thus, for example, the distance between two points A , with Cartesian coordinates ( a 1 , a 2 ) and B , with Cartesian coordinates ( b 1 , b 2 ) , can be defined by the expression d AB = s 2 ∑ i = 1 ( b i − a i ) 2 ≡ q ( b 1 − a 1 ) 2 +( b 2 − a 2 ) 2 . (1.2) In the n –dimensional space this distance is given by the expression d AB = s n ∑ i = 1 ( b i − a i ) 2 . (1.3) 1.3 Vector algebra In the previous section, construction of a coordinate system in two-dimensional space, which is intuitively close to human perception, was reviewed. In this system the distance between two points is measured by Pythagoras 2 formula (1.3). If, in such a space, a point is moved from position A to a new position B , this movement from A (start point) to B (end point) can be represented by the oriented straight line segment −→ AB (Fig. 1.3). | x ′ 1 | = OM ′ 1 OA , | x ′ 2 | = OM ′ 2 OB .

2 Π υϑαγ o ρας , Greek philosopher and mathematician. Born around 570 B.C. and died around 497 B.C. He is considered the founder of theoretical mathematics and research in physics (acoustics).