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and
a
·
b
cos
=
a
b
But what we don't know is the sense of ; that is, is
b
rotated in a counter-clockwise direction
relative to
a
, or vice versa? Well, consider the scenario shown in Fig. 2.34, where
b
is
rotated in
a counter-clockwise direction relative to
a
, creating a positive angle .
a
⊥
b
α
θ
a
Figure 2.34.
From the figure we can state that
=
a
⊥
a
⊥
·
b
b
cos
and
a
⊥
·
a
⊥
·
b
b
cos
=
=
(2.24)
a
⊥
b
a
b
But note that as
+
=
90
,
=
sin
cos
(2.25)
Substituting Eq. (2.25) in Eq. (2.24), we get
a
⊥
·
b
sin
=
(2.26)
a
b
Now, the nature of the cosine function is that
cos
=
cos
−
which is not the case for the sine function:
sin
−
=−
sin
