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Mathematics
Problem
Let let
z
1
,
z
2
,
z
3
z_1, z_2, z_3
z
1
,
z
2
,
z
3
be complex numbers of equal magnitude satisfying
z
1
+
z
2
+
z
3
=
−
3
2
−
i
5
z_1 + z_2 + z_3 = -\frac{\sqrt{3}}{2} - i\sqrt{5}
z
1
+
z
2
+
z
3
=
−
2
3
−
i
5
and
z
1
⋅
z
2
⋅
z
3
=
3
+
i
5
z_1 \cdot z_2 \cdot z_3 = \sqrt{3} + i\sqrt{5}
z
1
⋅
z
2
⋅
z
3
=
3
+
i
5
. If
z
1
=
x
1
+
i
y
1
,
z
2
=
x
2
+
i
y
2
z_1 = x_1 + i y_1, z_2 = x_2 + i y_2
z
1
=
x
1
+
i
y
1
,
z
2
=
x
2
+
i
y
2
and
z
3
=
x
3
+
i
y
3
;
x
1
,
x
2
,
x
3
,
y
1
,
y
2
,
y
3
∈
R
z_3 = x_3 + i y_3; x_1, x_2, x_3, y_1, y_2, y_3 \in \mathbb{R}
z
3
=
x
3
+
i
y
3
;
x
1
,
x
2
,
x
3
,
y
1
,
y
2
,
y
3
∈
R
, where
i
=
−
1
i = \sqrt{-1}
i
=
−
1
, then
16
(
x
1
y
1
+
x
2
y
2
+
x
3
y
3
)
2
5
=
?
\frac{16(x_1 y_1 + x_2 y_2 + x_3 y_3)^2}{5} = ?
5
16
(
x
1
y
1
+
x
2
y
2
+
x
3
y
3
)
2
=
?
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