By Henry G. Booker
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Extra resources for A Vector Approach to Oscillations
9(b). As an example let us consider the scalar quadratic equation i 2 + 2s + 5 = 0. 92). Consequently there is no scalar quantity s for which the expression on the left-hand side of Eq. 93) vanishes. This is illustrated in Fig. 19, which shows the graph of this function of s. 94) ί s 2 +2s+5 l· -1 0 1 FIG. 19. 93) has no scalar root. 5. Relation between Vector Algebra and Scalar Algebra 55 then the vector on the left-hand side vanishes for a pair of vectors s neither of which lie along the reference axis.
36) is illustrated in Fig. 7(a). Quadrature direction (a) FIG. 7. Illustrating calculation of the vector F from the vector s in accordance with Eq. 33). To evaluate for this vector the corresponding vector P given by Eq. 33) we have to add the vectors a0 and a^. The first of these is given by Eq. 34) and is illustrated in Fig. 7(b), while the second is obtained by performing the planar product of the vectors given in Eqs. 37) In Fig. 7(b) are illustrated the vectors a0 and a^ from Eqs. 3. 37) together with their vector sum, which evaluates to P = l2l§.
Algebraic Functions of a Vector 29 drawn from the origin O to the point labeled s. Using Fig. 1 we can now see that the vector corresponding to the factor s — Sj on the right-hand side of Eq. 4) is the vector running from the point sx to the point s in Fig. 2. Likewise the vector corresponding to the factor s — s 2 is the vector running from the point s2 to the point s, and so on. 6) therefore states that the vector P defined by Eq. 4) has a magnitude FIG. 2. Illustrating calculation of the vector P(s) given by Eq.