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

(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 vas in Fig. 3.3b; and added to uas in Fig. 3.3c – while uand vremain consistent with Fig. 3.3a. According to Eq. (3.99), the scalar product of wby itself reads

(3.100)

whereas combination with Eq. (3.55)leads to

(3.101)

after taking square roots of both sides, Eq. (3.101)becomes

(3.102)

where Eq. (3.99)was again invoked – so the length of a difference of vectors equals the square root of its scalar product by itself. In view of Eq. (3.70), one may rewrite Eq. (3.101)as

(3.103)

and a second application of the said distributive property conveys

(3.104)

insertion of Eq. (3.55)supports transformation to

(3.105)

whereas application of Eq. (3.58)further justifies

(3.106)

– also at the expense of the associative property of scalars. On the other hand, the definition of scalar product as per Eq. (3.53)permits reformulation of Eq. (3.106)to

(3.107)

thus retrieving Eq. (2.443)– as long as a ≡ ‖ u‖, b ≡ ‖ v‖, γ ≡ ∠ u,v, and c ≡ ‖ w‖. Equation (3.107)applies to the sides of the triangle in Fig. 3.3e, obtained, in turn, from that in Fig. 3.3d following counterclockwise rotation, so as to make ulie on the horizontal axis – followed by vertical and horizontal flipping (for graphical convenience). Remember that when uand vare normal to each other, the cosine of the angle between them is nil – so Eq. (3.107)would reduce to

(3.108)

under such circumstances; this is but Pythagoras’ theorem as per Eq. (2.431), with wplaying the role of hypotenuse, and uand vplaying the roles of sides of the right angle. This is illustrated in Fig. 3.3f in terms of sides uand v, with hypotenuse wgenerated in Fig. 3.3g as vector connecting the extreme points of uand v. The said theorem was proven previously based on Newton’s expansion of a difference, see Eqs. (2.432)and (2.433); it is possible to resort to a similar expansion of its conjugate, as illustrated in Fig. 3.3h. Two squares are accordingly considered therein – one with side a + b , and a smaller one with side c that is rotated as much as necessary to have its four corners simultaneously touch the sides of the original square; this originates four right triangles, all with hypotenuse c , and sides a and b . The area of the larger square is ( a + b ) 2, which may in turn be subdivided into the area of the smaller square, c 2, plus the areas of four identical triangles – each one accounting for ab /2, according to

(3.109)

expansion of the left‐hand side following Newton’s binomial, coupled with replacement of 4/2 by 2 in the right‐hand side, yields

(3.110)

which readily leads to Eq. (2.431)after dropping of 2 ab between sides.

Figure 3.3 Graphical representation of (a) vectors u and v , (b) u and symmetrical of v , denoted as vector −v , (c, d) sum of u and – v , denoted as (d) vector w, (d,e,g) with lengths ‖ u‖, ‖ v‖, and ‖ w‖, respectively, and (e) following rotation, and horizontal and vertical flipping of u, −v , and w; of (f) normal vectors uand vand (g) triangle with sides defined by uand v, and hypotenuse defined by w; and of (h) concentric squares, the larger with side a + b and the smaller with side c after rotation so as to touch the former at four points – with concomitant definition of lengths a and b .

The vector (or outer) product of two vectors is a third vector – denoted as u× v, and abiding to