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


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.