and this product can be expanded utilizing the distributive and commutative laws
to obtain
A·B =A 1 B 1 ˆe 1 ·eˆ 1 +A 1 B 2 ˆe 1 ·ˆe 2 +A 1 B 3 ˆe 1 ·ˆe 3
+A 2 B 1 ˆe 2 ·ˆe 1 +A 2 B 2 ˆe 2 ·ˆe 2 +A 2 B 3 ˆe 2 ·ˆe 3
+A 3 B 1 ˆe 3 ·ˆe 1 +A 3 B 2 ˆe 3 ·ˆe 2 +A 3 B 3 ˆe 3 ·ˆe 3.(6 .15)From the previous properties of the dot product of unit vectors, given by equations
(6.9), the dot product reduces to the form
A·B=A 1 B 1 +A 2 B 2 +A 3 B 3. (6 .16)Thus, the dot product of two vectors produces a scalar quantity which is the sum of
the products of like components.
From the definition of the dot product the following useful relationship results:
A·B =A 1 B 1 +A 2 B 2 +A 3 B 3 =|A||B|cos θ. (6 .17)This relation may be used to find the angle between two vectors when their origins
are made to coincide and their components are known. If in equation (6.17) one
makes the substitution A=B, there results the special formula
A·A=A^21 +A^22 +A^23 =A·Acos 0 = A^2 =|A|^2. (6 .18)Consequently, the magnitude of a vector Ais given by the square root of the sum of
the squares of its components or |A|=
√
A·A=√
A^21 +A^22 +A^23The previous dot product definition is moti-
vated by the law of cosines as the following argu-
ments demonstrate. Consider three points having
the coordinates (0, 0 ,0),(A 1 , A 2 , A 3 ), and (B 1 , B 2 , B 3 )
and plot these points in a cartesian coordinate sys-
tem as illustrated. Denote by A the directed line
segment from (0 , 0 ,0) to (A 1 , A 2 , A 3 ) and denote by
B the directed straight-line segment from (0 , 0 ,0) to
(B 1 , B 2 , B 3 ).
One can now apply the distance formula from analytic geometry to represent
the lengths of these line segments. We find these lengths can be represented by
|A|=√A^21 +A^22 +A^23 and |B|=
√
B 12 +B 22 +B^23.