– usually referred to as distributive property of scalar product of vectors, over vector addition on the right. The above graphical analysis emphasizes that the scalar product of two vectors is equivalent to the area of a rectangle, with one side defined by one such vectors and another side defined by the normal projection of the other vector onto the former; this is apparent in Fig. 3.2f for u · v, in Fig. 3.2g for u · w, and in Fig. 3.2h for u · ( v + w). The aforementioned distributive property is thus a consequence of the additivity of areas of juxtaposed rectangles – see Fig. 3.2h for area of rectangle representing u · ( v + w) and Fig. 3.2e for equivalent overall areas representing u · v and u · w . In view of the property conveyed by Eq. (3.58), one may also write
(3.71)
so combination with Eq. (3.70) transforms it to
a second application of the said commutative property allows transformation of Eq. (3.72) to
(3.73)
or, after renaming v, w and u as u, v and w, respectively,
– so the scalar product of vectors is also distributive over vector addition on the left.
Figure 3.2 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[0D], of straight segment [0D]; (c) projection of w onto u with magnitude equal to length, L[0E], of straight segment [0E]; (d) projection of v + w onto u with magnitude equal to length, L[0G], of straight segment [0G]; (e) sum of u · v, given by area, A[0AHI], of rectangle [0AHI], with u · w, given by area, A[HIJK], of rectangle [HIJK]; (f) scalar product, u · v, of u by v, given by area, A[0AHI], of rectangle [0AHI]; (g) scalar product, u · w, of u by w, given by area, A[0ALM], of rectangle [0ALM]; and (h) scalar product, u · ( v + w), of u by v + w, given by area, S[0AKJ], of rectangle [0AKJ].
Multiple products are also possible; consider first the scalar product of two vectors combined with the product of scalar by vector, say,
for which Eq. (3.53) was retrieved – with Eq. (2.2) assuring |s| = s, besides ∠s u, v = ∠ u, v when s is positive. Conversely, s < 0 implies |s| = − s also via Eq. (2.2), while ∠s u, v = π + ∠ u, v as the direction of su appears reversed relative to the original direction of u – thus implying cos{∠s u , v } = cos π cos {∠ u , v } − sin π sin {∠ u , v } as per Eq. (2.325), where cos π = −1 and sin π = 0 support, in turn, simplification to cos{∠s u , v } = − cos {∠ u , v }. Therefore, one would write
starting once more from Eq. (3.53). For conveying the same final result, Eqs. (3.75) and (3.76) can be condensed into the simpler version:
(3.77)
therefore, the dot product of the scalar multiple of a vector by another vector ends up being equal to the product of the said scalar by the dot product of the two vectors. A similar reasoning would allow one to write
at the expense of the algorithm labeled as Eq. (3.53), coupled with the commutative property of product of scalars; Eq. (3.78) is obviously equivalent to
(3.79)
after using Eq. (3.53) backward.
Since the scalar product of vector is itself a scalar, one may attempt to compute
(3.80)
stemming from Eq. (3.53); u may, in turn, appear as