Mathematics for Enzyme Reaction Kinetics and Reactor Performance. F. Xavier Malcata

in agreement with Fig. 3.4h and based on Eqs. (3.59), (3.112), and (3.121) – as represented in Fig. 3.4h. One may thus add the areas of the parallelograms in Figs. 3.4f and 3.4g to get

(3.123)

as emphasized in Fig. 3.4e via plain juxtaposition of the said parallelograms, with S_[BCKJ] = S_[0AHF]. However, triangles [0BJ] and [ACK] are identical as per Fig. 3.4i, owing to the geometrical features of parallelograms – i.e. [0B] and [AC] are parallel and share the same length, and the same applies to [BJ] and [CK]; hence, [0J] and [AK] must also be parallel, and have identical length. One accordingly concludes that

(3.124)

while Eqs. (3.118) and (3.120) allow transformation of Eq. (3.123) to

(3.125)

Recalling Eq. (3.111), one may redo Eq. (3.125) to

      (3.126)

      or else

(3.127)

upon addition and subtraction of S_[0BJ], and the aid of Eq. (3.124); the right-hand side of Eq. (3.127) is illustrated as [0AKJ] in Fig. 3.4i – which coincides in area with S_[0AIC] as in Fig. 3.4h, so one concludes that

(3.128)

following combination with Eqs. (3.111) and (3.122). Therefore, the vector product is distributive on the right with regard to addition of vectors. In what concerns the other possibility, Eq. (3.116) allows one to write

      (3.129)

where u + v plays here the role played previously by v, and similarly w plays here the role played previously by u; in view of Eq. (3.128), one gets

      (3.130)

      where Eq. (3.116) may again be invoked to obtain

(3.131)

      – so the vector product is distributive also on the right, with regard to addition of vectors.

Figure 3.4 Graphical representation of (a) vectors u, v, and w, and of sum, v + w, of v with w; (b) projection of v onto u^⊥, with magnitude equal to length, L_[BD], of straight segment [BD]; (c) projection of w onto u^⊥, with magnitude equal to length, L_[EF], of straight segment [EF]; (d) projection of v + w onto u^⊥ with magnitude equal to length, L_[CG], of straight segment [CG]; (e, i) sum of u × v, given by area, A_[0ACB], of parallelogram [0ACB], with u × w, given by area A_[BCKJ] of parallelogram [BCJK]; (f) vector product, u × v, of u by v, given by area, A_[0ACB], of parallelogram [0ACB]; (g) vector product, u × w, of u by w, given by area, A_[0AHF], of parallelogram [0AHF]; (h) vector product, u × ( v + w), of u by v + w, given by area, A_[0AIC], of parallelogram [0AIC]; and (i) equivalence of areas S_[0ACB] and S_[BCKJ] to area, S_[0AKJ], of parallelogram [0AKJ], via addition of area, S_[ACK], of triangle [ACK] and subtraction of area, S_[0BJ], of triangle [0BJ].

      Using the coordinate forms of vectors u and v as given by Eqs. (3.1) and (3.2), one can write

      (3.132)

Eq. (3.131) supports transformation to

      (3.133)

whereas Eq. (3.128) permits further transformation to

(3.134)