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

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_z is 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)

      where the distributive property of scalars allows transformation to

      (3.149)

      algebraic rearrangement at the expense of the commutative and associative properties of multiplication of scalars leads then to

      (3.150)

so Eqs. (3.22) and (3.51) may be invoked to write

      (3.151)

      that retrieves Eq. (3.128) after applying Eq. (3.143) twice – thus confirming validity of Eq. (3.128), through an independent derivation path.

Finally, it is worth mentioning that the volume, V, of a parallelepiped defined by vectors u, v, and w can be calculated as the area of the parallelogram that constitutes its base, defined by u and v and represented by vector (‖ u ‖‖ v ‖ sin {∠ u , v }) n as per Eq. (3.112) and (3.113), multiplied by its height – i.e. the projection of w upon n, and calculated as ‖ w ‖ cos {∠ w , n } as per Eq. (3.54). On the one hand, (‖ u ‖‖ v ‖ sin {∠ u , v }) n is, by definition, equal to u × v as per Eq. (3.111) – so ‖ u ‖‖ v ‖ sin {∠ u , v } coincides with ‖ u × v ‖ because ‖ n ‖ = 1; on the other hand, the length of the projection of w onto n reads ‖ w ‖ cos {∠ w , n }, where cos{∠ w , n } = cos {∠ w , u × v } since u × v has the direction of n . The product of ‖ u × v ‖ by ‖ w ‖ cos {∠ w , u × v } is but the scalar product of u × v by w as per Eq. (3.54) – so one finally concludes that