referring to Fig. 3.2f, as long as L[0D] = L[0I]; note that [0I] denotes a segment normal to [0A], with [0AHI] denoting a rectangle and S[0AHI] its area. By the same token, consider
(3.63)
as per Fig. 3.2c, with L[0F] denoting length of straight segment [0F] coinciding with w, such that its (orthogonal) projection over u looks like
– where [0E] denotes a straight segment in Fig. 3.2c, and [0M] denotes a straight segment in Fig. 3.2g that has the same length of [0E] but is normal to [0A]. Ordered multiplication of Eqs. (3.59) and (3.64) unfolds
where [0ALM] denotes the rectangle in Fig. 3.2g. Consider now the sum of v and w, as sketched in Fig. 3.2a – with length equal to L[0C], where [0C] denotes the straight segment coinciding with v + w; the (orthogonal) projection of v + w on u is given by
according to Fig. 3.2d, where straight segment [0G] is collinear with [0A] – and straight segment [0J] is perpendicular thereto, while sharing the same length with [0G], see Fig. 3.2h. Consequently,
based on Eqs. (3.59) and (3.66) – where rectangle [0AKJ] is laid out in Fig. 3.2h. Based on geometrical decomposition
– see Fig. 3.2 e–h; hence, one concludes that
(3.69)
stemming from Eqs. (3.62), (3.65), (3.67), and (3.68). In view of Eq. (3.53), one finally reaches
– 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
