one may redo Eq. (3.125) to
(3.126)
or else
upon addition and subtraction of S[0BJ], and the aid of Eq. (3.124); the right-hand side of Eq. (3.127) is illustrated as [0AKJ] in Fig. 3.4i – which coincides in area with S[0AIC] as in Fig. 3.4h, so one concludes that
following combination with Eqs. (3.111) and (3.122). Therefore, the vector product is distributive on the right with regard to addition of vectors. In what concerns the other possibility, Eq. (3.116) allows one to write
(3.129)
where u + v plays here the role played previously by v, and similarly w plays here the role played previously by u; in view of Eq. (3.128), one gets
(3.130)
where Eq. (3.116) may again be invoked to obtain
– so the vector product is distributive also on the right, with regard to addition of vectors.
Figure 3.4 Graphical representation of (a) vectors u, v, and w, and of sum, v + w, of v with w; (b) projection of v onto u⊥, with magnitude equal to length, L[BD], of straight segment [BD]; (c) projection of w onto u⊥, with magnitude equal to length, L[EF], of straight segment [EF]; (d) projection of v + w onto u⊥ with magnitude equal to length, L[CG], of straight segment [CG]; (e, i) sum of u × v, given by area, A[0ACB], of parallelogram [0ACB], with u × w, given by area A[BCKJ] of parallelogram [BCJK]; (f) vector product, u × v, of u by v, given by area, A[0ACB], of parallelogram [0ACB]; (g) vector product, u × w, of u by w, given by area, A[0AHF], of parallelogram [0AHF]; (h) vector product, u × ( v + w), of u by v + w, given by area, A[0AIC], of parallelogram [0AIC]; and (i) equivalence of areas S[0ACB] and S[BCKJ] to area, S[0AKJ], of parallelogram [0AKJ], via addition of area, S[ACK], of triangle [ACK] and subtraction of area, S[0BJ], of triangle [0BJ].
Using the coordinate forms of vectors u and v as given by Eqs. (3.1) and (3.2), one can write
(3.132)
Eq. (3.131) supports transformation to
(3.133)
whereas Eq. (3.128) permits further transformation to
In view of Eqs. (3.33) and (3.38), it is possible to reformulate Eq. (3.134) to
the algorithmic definition conveyed by Eq. (3.111) supports
because each of the vectors jx, jy, and jz is obviously collinear with itself – and the sine of a nil angle is nil. In addition,
(3.137)
(3.138)
and
(3.139)
– since the angle formed by each indicated pair of unit orthogonal vectors holds a unit sine, and the right‐hand‐sided mode is maintained; by the same token,
(3.140)
(3.141)
and
