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

the scalar product represents a length of vector u after multiplication by scaling factor ‖ v ‖ cos {∠ u , v }. As a consequence of Eq. (3.53), one has that

(3.55)

      because cos 0 is equal to unity. On the other hand, the definition provided by Eq. (3.53) implies that the scalar product is nil for two orthogonal vectors, i.e.

      (3.56)

– since the cosine of their angle is nil; hence, the scalar product being nil does not necessarily imply that at least one of the factors is a nil vector. In general, the scalar product of two collinear vectors is merely given by the product of their lengths – with Eq. (3.55) being a particular case of this statement.

      Since Eq. (3.53) may be rewritten as

      (3.57)

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

(3.58)

      after taking Eq. (3.53) into account – so the scalar product is itself commutative; note that the smaller angle formed by two vectors is not changed when their order is reversed.

The scalar product is distributive on the right with regard to addition – as graphically illustrated in Fig. 3.2. Consider first vector u as in Fig. 3.2a, with length given by

(3.59)

      where [0A] denotes a straight segment coinciding therewith – and likewise

      (3.60)

      with [0B] overlaid on v; the (orthogonal) projection of v on u will then exhibit length given by

(3.61)

where [0D] denotes a straight segment collinear with [0A], see Fig. 3.2b. In view of Eqs. (3.59) and (3.61), one realizes that

(3.62)

      referring to Fig. 3.2f, as long as L_[0D] = L_[0I]; note that [0I] denotes a segment normal to [0A], with [0AHI] denoting a rectangle and S_[0AHI] its area. By the same token, consider

      (3.63)

      as per Fig. 3.2c, with L_[0F] denoting length of straight segment [0F] coinciding with w, such that its (orthogonal) projection over u looks like

(3.64)

– where [0E] denotes a straight segment in Fig. 3.2c, and [0M] denotes a straight segment in Fig. 3.2g that has the same length of [0E] but is normal to [0A]. Ordered multiplication of Eqs. (3.59) and (3.64) unfolds

(3.65)

      where [0ALM] denotes the rectangle in Fig. 3.2g. Consider now the sum of v and w, as sketched in Fig. 3.2a – with length equal to L_[0C], where [0C] denotes the straight segment coinciding with v + w; the (orthogonal) projection of v + w on u is given by

(3.66)

      according to Fig. 3.2d, where straight segment [0G] is collinear with [0A] – and straight segment [0J] is perpendicular thereto, while sharing the same length with [0G], see Fig. 3.2h. Consequently,

(3.67)

based on Eqs. (3.59) and (3.66) – where rectangle [0AKJ] is laid out in Fig. 3.2h. Based on geometrical decomposition

(3.68)

      – see Fig. 3.2 e–h; hence, one concludes that

      (3.69)

stemming from Eqs. (3.62), (3.65), (3.67),