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
where ju denotes a unit vector colinear with u . Algebraic rearrangement resorting to Eq. (3.33) yields
(3.82)
from Eq. (3.81), whereas the associative property of multiplication of scalars unfolds
upon multiplication and division by cos{∠ v , w }, Eq. (3.83) becomes
with the aid also of the commutative property of multiplication of scalars. Recalling Eq. (3.53), one may reformulate Eq. (3.84) to
one promptly concludes that
because the vector in the right‐hand side of Eq. (3.85) has length equal to ‖ w ‖ multiplied by correction factor
(3.87)
this means that the scalar product of vectors is not associative with regard to the product of scalar by vector.
Although the definition as per Eq.
