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67. (a) We consider the following combinations:
∆
V
12
=
V
1
–
V
2
,
∆
V
13
=
V
1
–
V
3
, and
∆
V
23
=
V
2
–
V
3
. For
∆
V
12
,
∆
VA
t
A
t
A
t
At
dd
d
d
12
120
2
120
2
2
120
2
36
0
=−
−
°
=
°
F
H
G
I
K
J
−°
F
H
G
I
K
J
°
sin(
)
sin (
)
sin
cos
cos
ωω
ω
bg
where we use
( )
( )
sin
sin
2sin
2 cos
2
αβ
α
β
−=
−
+
⎡
⎤⎡
⎤
⎣
⎦⎣
⎦
and
sin 60
3 2.
°=
Similarly,
()
13
2
240
240
sin(
)
sin (
240 )
2 sin
cos
3 cos
120
22
d
d
t
t
A
t
A
A
t
°
⎛⎞
∆=
−
−
°
=
=
−
°
⎜⎟
⎝⎠
and
23
2
360
120
sin(
120 )
sin (
240 )
2 sin
cos
3 cos
180
d
d
t
t
A
t
A
°
−
−
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 Spring '08
 Any
 Physics

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