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

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(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 u and v labeled 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_z have 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

(3.97)

after recalling Eq. (3.19); algebraic manipulation transforms Eq. (3.97) to

      (3.98)

in view of the commutative and associative properties of addition of scalars; Eq. (3.95) may again be invoked to retrieve Eq. (3.70) with the aid of Eqs. (3.1) and (3.2), since u_x v_x + u_y v_y + u_z v_z = u · v and u_x w_x + u_y w_y + u_z w_z = u · w – and a similar reasoning would likewise generate Eq. (3.74).

      Equation (3.95) also leads to a number of other useful relationships; one of the most famous starts from vector w, defined as

(3.99)

according to Fig. 3.3d, after having obtained −v as symmetrical of vector v as in Fig. 3.3 b; and added to u as in Fig. 3.3c – while u and v remain consistent with Fig. 3.3a. According to Eq. (3.99), the scalar product of w by itself reads

      (3.100)

      whereas combination with Eq. (3.55) leads to

(3.101)

after taking square roots of both sides, Eq.

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