If either 
0 or
0,
a
b
=
=
r
r
r
r
 then θ is not defined, and in this case,
we define 
0
a b
 
r 
r
Observations
1 a b
⋅
r
r
 is a real number 2 Let 
aand
rb
r
be two nonzero vectors, then 
0
a b
⋅
r=
r
 if and only if 
aand
rb
r
 are
perpendicular to each other