rel="nofollow" href="#ulink_590d6c44-bf7e-5e00-b80a-709eae4f4def">Eq. (3.1) entails
(3.45)
which is equivalent to
(3.46)
due to Eq. (3.30); the distributive property of multiplication of scalars may again be invoked to write
(3.47)
whereas Eq. (3.19) justifies transformation to
Recalling Eq. (3.22), it is possible to convert Eq. (3.48) to
(3.49)
which can be combined with Eq. (3.30) to yield
(3.50)
Eq. (3.1) finally permits condensation to
thus proving that multiplication of scalar by vector is distributive also with regard to addition of scalars.
3.3 Scalar Multiplication of Vectors
The 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),
