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3.3 Scalar Multiplication of Vectors
ОглавлениеThe scalar (or inner) product of vectors – which may be represented by
is formally defined as
here ‖ u ‖ and ‖ v ‖ denote lengths of vectors u and v, respectively, and cos{∠ u , v } denotes cosine of (the smaller) angle formed by vectors u and v . If Eq. (3.53) is rewritten as
then the scalar product can be viewed as the product of the length of u by the length of the projection of v over u – see Eq. (2.288); in other words, the scalar product represents a length of vector u after multiplication by scaling factor ‖ v ‖ cos {∠ u , v }. As a consequence of Eq. (3.53), one has that
because cos 0 is equal to unity. On the other hand, the definition provided by Eq. (3.53) implies that the scalar product is nil for two orthogonal vectors, i.e.
(3.56)
– since the cosine of their angle is nil; hence, the scalar product being nil does not necessarily imply that at least one of the factors is a nil vector. In general, the scalar product of two collinear vectors is merely given by the product of their lengths – with Eq. (3.55) being a particular case of this statement.
Since Eq. (3.53) may be rewritten as
(3.57)
due to commutativity of the product of scalars, so one eventually finds that
after taking Eq. (3.53) into account – so the scalar product is itself commutative; note that the smaller angle formed by two vectors is not changed when their order is reversed.
The scalar product is distributive on the right with regard to addition – as graphically illustrated in Fig. 3.2. Consider first vector u as in Fig. 3.2a, with length given by
where [0A] denotes a straight segment coinciding therewith – and likewise
(3.60)
with [0B] overlaid on v; the (orthogonal) projection of v on u will then exhibit length given by
where [0D] denotes a straight segment collinear with [0A], see Fig. 3.2b. In view of Eqs. (3.59) and (3.61), one realizes that
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 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
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 rather than simply ‖ w ‖ as in Eq. (3.86) – and its direction is that of vector u (via ju), rather than that of vector w (or of unit vector jw, for that matter) also as in Eq. (3.86). Therefore, one may in general state that
(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. (3.53), or a graphical support (as done above) may be utilized to infer all properties of the scalar product of vectors, either approach may prove cumbersome in routine analysis – so a handier mode of calculation would be welcome. Toward this goal, one may resort to the coordinate‐based forms of vectors u and v labeled as Eqs. (3.1) and (3.2), i.e.
In view of Eq. (3.70), one can convert Eq. (3.88) to
(3.89)
and a further application of the said distributive property unfolds
Equations (3.33) and (3.38) permit transformation of Eq. (3.90) to
(3.91)
– or, due to Eq. (3.58),
Recalling Eq. (3.55), one realizes that
because vectors jx, jy, and jz have unit length by definition; on the other hand,
because each pair of indicated vectors are orthogonal to each other – so the cosine of their angle is nil, as per Eq. (3.56). Combination with Eqs. (3.93) and (3.94) permits simplification of Eq. (3.92) to just
or, in condensed form,
where i stands for x (i = 1), y (i = 2), or z (i = 3). Equation (3.95) is of particular relevance,since it allows calculation of the dot product based solely on the coordinates of its vector factors – without the need to explicitly know the angle between them or their magnitude; while providing a basic relationship of scalar product to the definition provided by Eq. (3.52) (as will soon be seen). Once in possession of Eq. (3.95), one realizes that
after recalling Eq. (3.19); algebraic manipulation transforms Eq. (3.97) to
(3.98)
in view of the commutative and associative properties of addition of scalars; Eq. (3.95) may again be invoked to retrieve Eq. (3.70) with the aid of Eqs. (3.1) and (3.2), since ux vx + uy vy + uz vz = u · v and ux wx + uy wy + uz wz = u · w – and a similar reasoning would likewise generate Eq. (3.74).
Equation (3.95) also leads to a number of other useful relationships; one of the most famous starts from vector w, defined as
according to Fig. 3.3d, after having obtained −v as symmetrical of vector v as in Fig. 3.3 b; and added to u as in Fig. 3.3c – while u and v remain consistent with Fig. 3.3a. According to Eq. (3.99), the scalar product of w by itself reads
(3.100)
whereas combination with Eq. (3.55) leads to
after taking square roots of both sides, Eq. (3.101) becomes
(3.102)
where Eq. (3.99) was again invoked – so the length of a difference of vectors equals the square root of its scalar product by itself. In view of Eq. (3.70), one may rewrite Eq. (3.101) as
(3.103)
and a second application of the said distributive property conveys
(3.104)
insertion of Eq. (3.55) supports transformation to
(3.105)
whereas application of Eq. (3.58) further justifies
– also at the expense of the associative property of scalars. On the other hand, the definition of scalar product as per Eq. (3.53) permits reformulation of Eq. (3.106) to
thus retrieving Eq. (2.443) – as long as a ≡ ‖ u ‖, b ≡ ‖ v ‖, γ ≡ ∠ u,v, and c ≡ ‖ w ‖. Equation (3.107) applies to the sides of the triangle in Fig. 3.3e, obtained, in turn, from that in Fig. 3.3d following counterclockwise rotation, so as to make u lie on the horizontal axis – followed by vertical and horizontal flipping (for graphical convenience). Remember that when u and v are normal to each other, the cosine of the angle between them is nil – so Eq. (3.107) would reduce to
(3.108)
under such circumstances; this is but Pythagoras’ theorem as per Eq. (2.431), with w playing the role of hypotenuse, and u and v playing the roles of sides of the right angle. This is illustrated in Fig. 3.3f in terms of sides u and v, with hypotenuse w generated in Fig. 3.3g as vector connecting the extreme points of u and v . The said theorem was proven previously based on Newton’s expansion of a difference, see Eqs. (2.432) and (2.433); it is possible to resort to a similar expansion of its conjugate, as illustrated in Fig. 3.3h. Two squares are accordingly considered therein – one with side a + b, and a smaller one with side c that is rotated as much as necessary to have its four corners simultaneously touch the sides of the original square; this originates four right triangles, all with hypotenuse c, and sides a and b. The area of the larger square is (a + b)2, which may in turn be subdivided into the area of the smaller square, c2, plus the areas of four identical triangles – each one accounting for ab/2, according to
(3.109)
expansion of the left‐hand side following Newton’s binomial, coupled with replacement of 4/2 by 2 in the right‐hand side, yields
(3.110)
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.
