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

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(3.8), after taking square roots of both sides – as long as ∣u_x ∣ ≡ ‖ u_x ‖, ∣u_y ∣ ≡ ‖ u_y ‖, and ∣u_z ∣ ≡ ‖ u_z ‖; a similar reasoning obviously applies to v_x, v_y, and v_z describing v . The general rules of trigonometry indicate, in turn, that angle θ_u (or θ_v) – formed with the y‐axis by the projection of u (or v) onto the y0z plane, is such that

(3.13)

      and likewise

      (3.14)

similar expressions can be laid out pertaining to angle ϕ_u (or ϕ_v) formed with the z‐axis by the projection of u (or v) onto the x0z plane, viz.

      (3.15)

      and similarly

(3.16)

Therefore, equality of u and v requires that ‖ u ‖ = ‖ v ‖ describing the same length, and θ_u = θ_v and ϕ_u = ϕ_v describing the same orientation – thus using up all three spatial degrees of freedom; upon inspection of the functional forms of Eqs. (3.8), (3.9), and (3.13)–(3.16), one concludes that the said three equalities unequivocally enforce u_x = v_x, u_y = v_y, and u_z = v_z .

3.1 Addition of Vectors

The simpler operation involving vectors is addition; to be applied to u and v, the point of origin of v must be made coincident with the point of termination of u . The vector sum, u + v, is thus the vector with the same point of origin of u and the same point of termination of v, as apparent in Fig. 3.1a. Therefore, one finds in general

      (3.17)

with the equal sign holding when u and v are collinear with the same orientation; coupled with

      (3.18)

      with the equal sign holding again when u and v are collinear and point in the same direction. Here ∠ u, v and ∠ u, u + v denote the angles formed by vectors u and v, and by vectors u and u + v, respectively. Since such coordinates are simply the normal projection of the vector at stake onto the corresponding Cartesian axes, addition of two vectors corresponds to merely adding the homologous coordinates, i.e.

(3.19)

based on Eqs. (3.1) and (3.2); both these statements are apparent from inspection of Fig. 3.1a.

Figure 3.1 Graphical representation of (a) addition of two vectors, u and v, and (b) multiplication of scalar α by vector u .

Addition of vectors is commutative – since, according to Eq. (3.19),

      (3.20)

      this is equivalent to

(3.21)

      as per the commutative property of addition of scalars – so

(3.22)

following combination of Eqs. (3.19) and (3.21).

      Given a third vector w – defined as

(3.23)

      one may recall Eqs. (3.1), (3.2), and (3.19) to write

(3.24)

because of the commutative property of vector addition as per Eq. (3.22), one can rewrite Eq. (3.24) as

      (3.25)

      whereas Eq. (3.19) leads to

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