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

inspection of Eqs. (3.30) and (3.33) – so multiplication of scalar by vector is commutative.

      Denoting a second scalar by β, it can be stated that

      (3.34)

      with the aid of Eq. (3.30); a second application of the algorithm conveyed by Eq. (3.30) unfolds

      (3.35)

      – where the associative property of multiplication of scalars supports

(3.36)

Equation (3.36) may be rewritten as

      (3.37)

      at the expense of Eq. (3.30), which condenses to

(3.38)

      with the aid of Eq. (3.1); this means that multiplication of scalar by vector is associative.

One finally realizes that

      (3.39)

      in view of Eq. (3.19), which becomes

      (3.40)

      as per Eq. (3.30); the distributive property of multiplication of scalars has it that

      (3.41)

      where Eq. (3.19) taken backward supports conversion to

(3.42)

Equation (3.22) finally permits transformation of Eq. (3.42) to

      (3.43)

      which prompts

      (3.44)

      once Eqs. (3.1) and (3.2) are recalled; hence, multiplication of scalar by vector is distributive with regard to addition of vectors.

      On the other hand, Eq. (3.1) entails

      (3.45)

      which is equivalent to

      (3.46)

      due to Eq. (3.30); the distributive property of multiplication of scalars may again be invoked to write

      (3.47)

      whereas Eq. (3.19) justifies transformation to

(3.48)

Recalling Eq. (3.22), it is possible to convert Eq. (3.48) to

      (3.49)

      which can be combined with Eq. (3.30) to yield

      (3.50)

      Eq. (3.1) finally permits condensation to

(3.51)

      thus proving that multiplication of scalar by vector is distributive also with regard to addition of scalars.

3.3 Scalar Multiplication of Vectors

      The scalar (or inner) product of vectors – which may be represented by

(3.52)

      is formally defined as

(3.53)

here ‖ u ‖ and ‖ v ‖ denote lengths of vectors u and v, respectively, and cos{∠ u , v } denotes cosine of (the smaller) angle formed by vectors u and v . If Eq. (3.53) is rewritten as

(3.54)

      then the scalar product can be viewed as the product of the length of u by the length of the projection of v over u – see Eq.