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Mathematics
Problem
If
z
1
=
a
+
i
b
z_1 = a + ib
z
1
=
a
+
ib
and
z
2
=
c
+
i
d
z_2 = c + id
z
2
=
c
+
i
d
are complex numbers such that
∣
z
1
∣
=
∣
z
2
∣
=
1
|z_1| = |z_2| = 1
∣
z
1
∣
=
∣
z
2
∣
=
1
and
Re
(
z
1
z
ˉ
2
)
=
0
\operatorname{Re}(z_1 \bar{z}_2) = 0
Re
(
z
1
z
ˉ
2
)
=
0
, where
i
=
−
1
i = \sqrt{-1}
i
=
−
1
, then the complex numbers
ω
1
=
a
+
i
c
\omega_1 = a + ic
ω
1
=
a
+
i
c
and
ω
2
=
b
+
i
d
\omega_2 = b + id
ω
2
=
b
+
i
d
satisfy a.
∣
ω
1
∣
=
1
|\omega_1| = 1
∣
ω
1
∣
=
1
b.
∣
ω
2
∣
=
1
|\omega_2| = 1
∣
ω
2
∣
=
1
c.
Re
(
ω
1
ω
ˉ
2
)
=
0
\operatorname{Re}(\omega_1 \bar{\omega}_2) = 0
Re
(
ω
1
ω
ˉ
2
)
=
0
d. None of these
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