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

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.

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

(3.54)

then the scalar product can be viewed as the product of the length of uby the length of the projection of vover u– see Eq. (2.288); in other words, the scalar product represents a length of vector uafter 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 uas in Fig. 3.2a, with length given by

(3.59)

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

(3.60)

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

(3.61)