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

(3.76)

starting once more from Eq. (3.53). For conveying the same final result, Eqs. (3.75)and (3.76)can be condensed into the simpler version:

(3.77)

therefore, the dot product of the scalar multiple of a vector by another vector ends up being equal to the product of the said scalar by the dot product of the two vectors. A similar reasoning would allow one to write

(3.78)

at the expense of the algorithm labeled as Eq. (3.53), coupled with the commutative property of product of scalars; Eq. (3.78)is obviously equivalent to

(3.79)

after using Eq. (3.53)backward.

Since the scalar product of vector is itself a scalar, one may attempt to compute

(3.80)

stemming from Eq. (3.53); umay, in turn, appear as

(3.81)

where j udenotes a unit vector colinear with u. Algebraic rearrangement resorting to Eq. (3.33)yields

(3.82)

from Eq. (3.81), whereas the associative property of multiplication of scalars unfolds

(3.83)

upon multiplication and division by cos{∠ v, w}, Eq. (3.83)becomes

(3.84)

with the aid also of the commutative property of multiplication of scalars. Recalling Eq. (3.53), one may reformulate Eq. (3.84)to

(3.85)

one promptly concludes that

(3.86)

because the vector in the right‐hand side of Eq. (3.85)has length equal to ‖ w‖ multiplied by correction factor rather than simply ‖ w‖ as in Eq. (3.86)– and its direction is that of vector u(via j u), rather than that of vector w(or of unit vector j w, for that matter) also as in Eq. (3.86). Therefore, one may in general state that

(3.87)

this means that the scalar product of vectors is not associative with regard to the product of scalar by vector.

Although the definition as per Eq. (3.53), or a graphical support (as done above) may be utilized to infer all properties of the scalar product of vectors, either approach may prove cumbersome in routine analysis – so a handier mode of calculation would be welcome. Toward this goal, one may resort to the coordinate‐based forms of vectors uand vlabeled as Eqs. (3.1)and (3.2), i.e.

(3.88)

In view of Eq. (3.70), one can convert Eq. (3.88)to

(3.89)

and a further application of the said distributive property unfolds

(3.90)

Equations (3.33)and (3.38)permit transformation of Eq. (3.90)to

(3.91)

– or, due to Eq. (3.58),

(3.92)

Recalling Eq. (3.55), one realizes that

(3.93)

because vectors j x, j y, and j zhave unit length by definition; on the other hand,

(3.94)

because each pair of indicated vectors are orthogonal to each other – so the cosine of their angle is nil, as per Eq. (3.56). Combination with Eqs. (3.93)and (3.94)permits simplification of Eq. (3.92)to just

(3.95)

or, in condensed form,

(3.96)

where i stands for x ( i = 1), y ( i = 2), or z ( i = 3). Equation (3.95)is of particular relevance,since it allows calculation of the dot product based solely on the coordinates of its vector factors – without the need to explicitly know the angle between them or their magnitude; while providing a basic relationship of scalar product to the definition provided by Eq. (3.52)(as will soon be seen). Once in possession of Eq. (3.95), one realizes that