Mathematical Physics Vol 1

Chapter 1. Vector algebra

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said that a coordinate system with mutually perpendicular axes is called the Cartesian coordinate system. It is common to denote the axes of the Cartesian coordinate system by x , y and z , instead of x 1 , x 2 and x 3 , respectively, and the corresponding base vectors by i , j and k , instead of e 1 , e 2 and e 3 , respectively. Note that both the left and the right coordinate systems are used, although the right one is more common (Fig. 1.19).

z

i

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k j

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(a) Right orientation.

(b) Left orientation.

Figure 1.19: Orientation of coordinate systems. Consider now an arbitrary vector a , represented by the oriented segment −→ AB , where A is the start, and B the end of segment AB (Fig. 1.20).

Figure 1.20: Projection of vectors expressed in terms of coordinates of their start and end points.

If two points A ( x A , y A , z A ) and B ( x B , y B , z B ) are given by their coordinates, and the vectors r A and r B are the position vectors of these points, then: −→ AB = a = r B − r A ⇒ a x = x B − x A , a y = y B − y A i a z = z B − z A (1.37) where a x , a y and a z are measures of the vector a with respect to the coordinate system, which is shortly denoted, for simplicity, by a =[ a x , a y , a z ] (1.38) instead by a = a x i + a y j + a z k . Let us now express the previously defined concepts through the corresponding measures. The magnitude of a vector a is, by its definition, the distance between points A and B , which, according to (1.3), can be represented in the Euclidean space by the relation | a | = q ( x B − x A ) 2 +( y B − y A ) 2 +( z B − z A ) 2 = q a 2 x + a 2 y + a 2 z (1.39)