ABSTRACT
The scalar product of two vectors a and b is a form of multiplication of two vectors which yields a scalar quantity.
Let a and b be any two vectors, and let u be the angle between them when
their initial points are brought into coincidence, with 00/u0/p , as shown in Fig. 131. The scalar product, or dot product as it is also called, of a and b is
written a b and defined as
a b½a½ ½b½ cos u:
The definition shows that the scalar product is commutative because
a bb a:
It follows directly from the above definition that if a and b are orthogonal
(perpendicular), u/p /2 and so
a b0:
If, however, a and b are parallel (u/0), then
a b ½a½ ½b½
and, in particular,
a aa2 ½a½2:
If the line segments of two vectors a and b are parallel, but the senses
(directions of the arrows) are opposite, the vectors are sometimes said to be anti-parallel. In this case u/p and it follows that then
a b½a½ ½b½:
It follows from these results that
while
i j j i i kk i j kk j0:
The scalar product also obeys the distributive law
a (bc)a ba c:
This last result is used to evaluate scalar products when a and b are given in
their Cartesian form. If aa1ia2ja3k and bb1ib2jb3k, we have
a b(a1ia2ja3k) (b1ib2jb3k),
and an application of the distributive law coupled with the results for scalar
products involving i, j and k gives the fundamental result
a ba1b1a2b2a3b3:
Equating the two definitions of a b gives
½a½ ½b½ cos ua b,
so the angle u between a and b follows from
cos u a b ½a½ ½b½
:
This last result is seen to be equivalent to
cos u aˆ bˆ,
so the cosine of the angle between a and b equals the scalar product of the
unit vectors aˆ and bˆ along a and b, respectively.