Vectors

2 A vector is a directed line segment which can denote a variety of quantities, such as position of point with respect to another (position vector), a force, or a traction.

3 A vector may be defined with respect to a particular coordinate system by specifying the components of the vector in that system. The choice of the coordinate system is arbitrary, but some are more suitable than others (axes corresponding to the major direction of the object being analyzed).

4 The rectangular Cartesian coordinate system is the most often used one (others are the cylindrical, spherical or curvilinear systems). The rectangular system is often represented by three mutually perpendicular axes Oxyz, with corresponding unit vector triad i, j, k (or ei, e2, e3) such that:

ii = jj = k-k =1 (1.1-b) ij = j-k = k-i = 0 (1.1-c)

Such a set of base vectors constitutes an orthonormal basis.

5 An arbitrary vector v may be expressed by v = vxi + Vyj + vz k (1.2)

are the projections of v onto the coordinate axes, Fig. 1.1.

Figure 1.1: Direction Cosines (to be corrected)

Figure 1.1: Direction Cosines (to be corrected)

The unit vector in the direction of v is given by e^ = — = cos ai + cos ßj + cos 7k

Since v is arbitrary, it follows that any unit vector will have direction cosines of that vector as its Cartesian components.

7 The length or more precisely the magnitude of the vector is denoted by || v ||= \Jvf + v\ + v|.

8 We will denote the contravariant components of a vector by superscripts vk, and its covariant components by subscripts vk (the significance of those terms will be clarified in Sect. 1.1.2.1.

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