which readily leads to Eq. (2.431) after dropping of 2ab 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 u and v and (g) triangle with sides defined by u and 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.
3.4 Vector Multiplication of Vectors
The vector (or outer) product of two vectors is a third vector – denoted as u × v, and abiding to
here sin{∠ u , v } denotes sine of (the smaller) angle formed by vectors u and v – and n denotes unit vector normal to the plane containing u and v, and oriented such that u, v, and n form a right‐handed system. As will be proven in due time, the area, S, of a parallelogram with sides accounted for by u and v is given by the product of its base, ‖ u ‖, by its heigth – which is, in turn, obtained as the projection of v onto u⊥, i.e. ‖ v ‖ sin {∠ u , v }, as given by
hence, Eq. (3.111) can be rewritten as
(3.113)
meaning that the vector product defines the vector area, Sn, of the portion of plane bounded by vectors u and v . The definition conveyed by Eq. (3.111) implies that the vector product is nil for two collinear vectors, because the sine of the angle formed thereby is nil; hence, the vector product being nil does not necessarily imply that at least one of the factors is a nil vector itself.
The vector product is not commutative; in fact,
stemming from Eq. (3.111), where −n appears because the vector system is now left handed; Eq. (3.114) may thus be rewritten as
(3.115)
due to commutativity of the product of scalars, so one eventually finds
– which means that the vector product is actually anticommutative.
Consider now vectors u, v, and w as depicted in Fig. 3.4a. Equation (3.59) still holds, relating ‖ u ‖ to L[0A], as well as Eq. (3.111) pertaining to u × v, while one has that
as per Fig. 3.4b – where [BD], with length L[BD], denotes a straight segment opposed to ∠ u, v and obtained after projection of v onto u⊥. Consequently,
based on Eqs. (3.59), (3.112), and (3.117) – and illustrated in Fig. 3.4f. By the same token,
as per Fig. 3.4c, where [EF] denotes a straight segment opposed to ∠ u, w and obtained via projection of w onto u⊥; therefore,
stemming from Eqs. (3.59), (3.112), and (3.119) – and apparent in Fig. 3.4g. Finally,
as per Fig. 3.4d, where [CG] denotes a straight segment opposed to ∠ u, v + w and generated through projection of v + w onto u⊥; hence,