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

(3.111)

here sin{∠ u, v} denotes sine of (the smaller) angle formed by vectors uand v– and ndenotes unit vector normal to the plane containing uand v, and oriented such that u, v, and nform a right‐handed system. As will be proven in due time, the area, S , of a parallelogram with sides accounted for by uand vis given by the product of its base, ‖ u‖, by its heigth – which is, in turn, obtained as the projection of vonto u ⊥, i.e. ‖ v‖ sin {∠ u, v}, as given by

(3.112)

hence, Eq. (3.111)can be rewritten as

(3.113)

meaning that the vector product defines the vector area, Sn , of the portion of plane bounded by vectors uand v. The definition conveyed by Eq. (3.111)implies that the vector product is nil for two collinear vectors, because the sine of the angle formed thereby is nil; hence, the vector product being nil does not necessarily imply that at least one of the factors is a nil vector itself.

The vector product is not commutative; in fact,

(3.114)

stemming from Eq. (3.111), where −n appears because the vector system is now left handed; Eq. (3.114)may thus be rewritten as

(3.115)

due to commutativity of the product of scalars, so one eventually finds

(3.116)

– which means that the vector product is actually anticommutative.

Consider now vectors u, v, and was depicted in Fig. 3.4a. Equation (3.59)still holds, relating ‖ u‖ to L [0A], as well as Eq. (3.111)pertaining to u× v, while one has that

(3.117)

as per Fig. 3.4b – where [ BD ], with length L [BD], denotes a straight segment opposed to ∠ u, vand obtained after projection of vonto u ⊥. Consequently,

(3.118)

based on Eqs. (3.59), (3.112), and (3.117)– and illustrated in Fig. 3.4f. By the same token,

(3.119)

as per Fig. 3.4c, where [ EF ] denotes a straight segment opposed to ∠ u, wand obtained via projection of wonto u ⊥; therefore,

(3.120)

stemming from Eqs. (3.59), (3.112), and (3.119)– and apparent in Fig. 3.4g. Finally,

(3.121)

as per Fig. 3.4d, where [ CG ] denotes a straight segment opposed to ∠ u, v+ wand generated through projection of v + w onto u ⊥; hence,

(3.122)

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 [0 BJ ] and [ ACK ] are identical as per Fig. 3.4i, owing to the geometrical features of parallelograms – i.e. [0 B ] and [ AC ] are parallel and share the same length, and the same applies to [ BJ ] and [ CK ]; hence, [0 J ] 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 [0 AKJ ] 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