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

(3.17)

with the equal sign holding when uand vare collinear with the same orientation; coupled with

(3.18)

with the equal sign holding again when uand vare collinear and point in the same direction. Here ∠ u, vand ∠ u, u+ vdenote the angles formed by vectors uand v, and by vectors uand u + v , respectively. Since such coordinates are simply the normal projection of the vector at stake onto the corresponding Cartesian axes, addition of two vectors corresponds to merely adding the homologous coordinates, i.e.

(3.19)

based on Eqs. (3.1)and (3.2); both these statements are apparent from inspection of Fig. 3.1a.

Figure 3.1 Graphical representation of (a) addition of two vectors, uand v, and (b) multiplication of scalar α by vector u.

Addition of vectors is commutative – since, according to Eq. (3.19),

(3.20)

this is equivalent to

(3.21)

as per the commutative property of addition of scalars – so

(3.22)

following combination of Eqs. (3.19)and (3.21).

Given a third vector w– defined as

(3.23)

one may recall Eqs. (3.1), (3.2), and (3.19)to write

(3.24)

because of the commutative property of vector addition as per Eq. (3.22), one can rewrite Eq. (3.24)as

(3.25)

whereas Eq. (3.19)leads to

(3.26)

with the aid of Eqs. (3.1), (3.2), and (3.23). Algebraic rearrangement of Eq. (3.26)– at the expense again of Eq. (3.19), produces

(3.27)

which leads directly to

(3.28)

due to Eq. (3.22); one finally attains

(3.29)

in view again of Eqs. (3.19)and (3.23)– so addition of vectors is associative.

Another common operation is multiplication of vector uby scalar α ; this produces a new vector αu , collinear with ubut with opposite direction if α < 0 – with length given by | α |‖ u‖, as apparent in Fig. 3.1b. Using vector coordinates, one accordingly finds that

(3.30)

– so the coordinates in each direction of space are expanded (or contracted) proportionally. Based on Eq. (3.30), one may equivalently write

(3.31)

due to the commutativity of the product of scalars, Eq. (3.31)yields

(3.32)

One therefore concludes that

(3.33)

following comparative 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.