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

(3.111)

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

(3.112)

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,

(3.114)

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

(3.116)

– 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

(3.117)

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,

(3.118)

based on Eqs. (3.59), (3.112), and (3.117) – and illustrated in Fig. 3.4f. By the same token,

(3.119)

as per Fig. 3.4c, where [EF] denotes a straight segment opposed to ∠ u, w and obtained via projection of w onto u^⊥; therefore,

(3.120)

stemming from Eqs. (3.59), (3.112), and (3.119) – and apparent in Fig. 3.4g. Finally,

(3.121)

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,

(3.122)

in agreement with Fig. 3.4h and based on Eqs. (3.59), (3.112), and (3.121) – as represented in Fig. 3.4h. One may thus add the areas of the parallelograms in Figs. 3.4f and 3.4g to get

(3.123)

as emphasized in Fig. 3.4e via plain juxtaposition of the said parallelograms, with S_[BCKJ] = S_[0AHF]. However, triangles [0BJ] and [ACK] are identical as per Fig. 3.4i, owing to the geometrical features of parallelograms – i.e. [0B] and [AC] are parallel and share the same length, and the same applies to [BJ] and [CK]; hence, [0J] and [AK] must also be parallel, and have identical length. One accordingly concludes that

(3.124)

while Eqs. (3.118) and (3.120) allow transformation of Eq. (3.123) to

(3.125)

Recalling Eq. (3.111), one may redo Eq. (3.125) to

(3.126)

or else

(3.127)

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

(3.128)

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

(3.131)

– 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

(3.134)

In view of Eqs. (3.33) and (3.38), it is possible to reformulate Eq. (3.134) to

(3.135)

the algorithmic definition conveyed by Eq. (3.111) supports

(3.136)

because each of the vectors j_x, j_y, and j_z 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

(3.142)

because a left‐hand‐sided vector system would result in this case. Upon insertion of Eqs. (3.136)–(3.142), it becomes possible to redo Eq. (3.135) to

(3.143)

together with factoring out of j_x, j_y, and j_z; the resulting vector, u × v, may appear in the alternative form

(3.144)

resorting to matrix notation, or equivalently

(3.145)

at the expense of the concept of determinant (both to be introduced later), combined with Eq. (1.9). One may instead write

(3.146)

– as alias of Eq. (3.143); similarly to Eq. (3.96), i stands for x (i = 1), y (i = 2), or z (i = 3), should the alternating operator, δ_ijk, be defined by

(3.147)

Once in possession of Eqs. (3.19) and (3.143), one may revisit Eq. (3.128) as

(3.148)

where the distributive property of scalars allows transformation to

(3.149)

algebraic rearrangement at the expense of the commutative and associative properties of multiplication of scalars leads then to

(3.150)

so Eqs. (3.22) and (3.51) may be invoked to write

(3.151)

that retrieves Eq. (3.128) after applying Eq. (3.143) twice – thus confirming validity of Eq. (3.128), through an independent derivation path.

Finally, it is worth mentioning that the volume, V, of a parallelepiped defined by vectors u, v, and w can be calculated as the area of the parallelogram that constitutes its base, defined by u and v and represented by vector (‖ u ‖‖ v ‖ sin {∠ u , v }) n as per Eq. (3.112) and (3.113), multiplied by its height – i.e. the projection of w upon n, and calculated as ‖ w ‖ cos {∠ w , n } as per Eq. (3.54). On the one hand, (‖ u ‖‖ v ‖ sin {∠ u , v }) n is, by definition, equal to u × v as per Eq. (3.111) – so ‖ u ‖‖ v ‖ sin {∠ u , v } coincides with ‖ u × v ‖ because ‖ n ‖ = 1; on the other hand, the length of the projection of w onto n reads ‖ w ‖ cos {∠ w , n }, where cos{∠ w , n } = cos {∠ w , u × v } since u × v has the direction of n . The product of ‖ u × v ‖ by ‖ w ‖ cos {∠ w , u × v } is but the scalar product of u × v by w as per Eq. (3.54) – so one finally concludes that

(3.152)

with a scalar quantity being now at stake.