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

(3.129)

where u + v plays here the role played previously by v, and similarly wplays 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 vwith w; (b) projection of vonto u ⊥, with magnitude equal to length, L [BD], of straight segment [ BD ]; (c) projection of wonto 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 [0 ACB ], with u× w, given by area A [BCKJ]of parallelogram [ BCJK ]; (f) vector product, u× v, of uby v, given by area, A [0ACB], of parallelogram [0 ACB ]; (g) vector product, u× w, of uby w, given by area, A [0AHF], of parallelogram [0 AHF ]; (h) vector product, u× ( v + w ), of uby v + w , given by area, A [0AIC], of parallelogram [0 AIC ]; and (i) equivalence of areas S [0ACB]and S [BCKJ]to area, S [0AKJ], of parallelogram [0 AKJ ], via addition of area, S [ACK], of triangle [ ACK ] and subtraction of area, S [0BJ], of triangle [0 BJ ].

Using the coordinate forms of vectors uand vas 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)

In view of Eqs. (3.33)and (3.38), it is possible to reformulate Eq. (3.134)to

(3.135)

the algorithmic definition conveyed by Eq. (3.111)supports

(3.136)

because each of the vectors j x, j y, and j zis obviously collinear with itself – and the sine of a nil angle is nil. In addition,

(3.137)

(3.138)

and

(3.139)

– since the angle formed by each indicated pair of unit orthogonal vectors holds a unit sine, and the right‐hand‐sided mode is maintained; by the same token,

(3.140)

(3.141)

and

(3.142)

because a left‐hand‐sided vector system would result in this case. Upon insertion of Eqs. (3.136)– (3.142), it becomes possible to redo Eq. (3.135)to

(3.143)

together with factoring out of j x, j y, and j z; the resulting vector, u× v, may appear in the alternative form

(3.144)

resorting to matrix notation, or equivalently

(3.145)

at the expense of the concept of determinant (both to be introduced later), combined with Eq. (1.9). One may instead write

(3.146)

– as alias of Eq. (3.143); similarly to Eq. (3.96), i stands for x ( i = 1), y ( i = 2), or z ( i = 3), should the alternating operator, δ ijk, be defined by

(3.147)

Once in possession of Eqs. (3.19)and (3.143), one may revisit Eq. (3.128)as

(3.148)