There exist two complex numbers $c$, say $c_1$ and $c_2$, so that $3 + 2i$, $6 + i$, and $c$ form the vertices of an equilateral triangle. Find the product $c_1 c_2$ in rectangular form.
[asy]
import TrigMacros; 
size(200); 

pair A, B;
pair[] C;

rr_cartesian_axes(-2, 8, -4, 7, complexplane = true); 
A = (3,2);
B = (6,1);
C[1] = rotate(60,A)*(B);
C[2] = rotate(60,B)*(A);

draw(A--C[1]--B--C[2]--cycle);
draw(A--B);

dot("$3 + 2i$", A, 2*NNW);
dot("$6 + i$", B, E);
dot("$c_1$", C[1], N);
dot("$c_2$", C[2], S);
[/asy]
Hint(s):
Solve for $c_1$ and $c_2$ using exponential form, but don't simplify them until you do the multiplication: it simplifies calculations.

We are given two complex numbers, $ 3 + 2i $ and $ 6 + i $, which represent two vertices of an equilateral triangle. We need to find the product of the two possible complex numbers $ c_1 $ and $ c_2 $, which represent the third vertex of the equilateral triangle.

### Step 1: Representing complex numbers geometrically
Let the complex numbers $ z_1 = 3 + 2i $ and $ z_2 = 6 + i $ be two points in the complex plane corresponding to two vertices of the equilateral triangle. The third vertex, $ c $, could either be $ c_1 $ or $ c_2 $, as there are two possible locations for this point.

### Step 2: Use rotation in the complex plane
To find the third vertex of the equilateral triangle, we use the fact that rotating one of the points by $ 60^\circ $ around the other point gives us one of the possible third vertices. Specifically, rotating $ z_2 $ around $ z_1 $ by $ 60^\circ $ gives one vertex, and rotating $ z_2 $ by $ -60^\circ $ gives the other vertex.

The formula for rotating a point $ z $ by an angle $ \theta $ around another point $ z_0 $ is:
\[
z' = z_0 + (z - z_0) e^{i \theta}
\]
For our case, $ \theta = 60^\circ = \frac{\pi}{3} $ or $ \theta = -60^\circ = -\frac{\pi}{3} $.

### Step 3: Compute the two possible third vertices
We will compute the two possible third vertices $ c_1 $ and $ c_2 $.

1. **For $ c_1 $** (rotating by $ 60^\circ $):
   \[
   c_1 = z_1 + (z_2 - z_1) e^{i \frac{\pi}{3}}
   \]
   Here, $ z_2 - z_1 = (6 + i) - (3 + 2i) = 3 - i $. Thus:
   \[
   c_1 = (3 + 2i) + (3 - i) e^{i \frac{\pi}{3}}
   \]

2. **For $ c_2 $** (rotating by $ -60^\circ $):
   \[
   c_2 = z_1 + (z_2 - z_1) e^{-i \frac{\pi}{3}}
   \]
   Similarly:
   \[
   c_2 = (3 + 2i) + (3 - i) e^{-i \frac{\pi}{3}}
   \]

### Step 4: Multiply $ c_1 $ and $ c_2 $
Rather than simplifying $ c_1 $ and $ c_2 $ individually, we can directly compute the product. Using the fact that the product of two complex numbers involves multiplying their magnitudes and adding their arguments, and noting that the arguments $ \frac{\pi}{3} $ and $ -\frac{\pi}{3} $ cancel each other out, we have:
\[
c_1 c_2 = \left[ (3 + 2i) + (3 - i) e^{i \frac{\pi}{3}} \right] \left[ (3 + 2i) + (3 - i) e^{-i \frac{\pi}{3}} \right]
\]
This expression simplifies nicely using the identity $ e^{i \frac{\pi}{3}} \cdot e^{-i \frac{\pi}{3}} = 1 $.

Thus, the product $ c_1 c_2 $ simplifies to:
\[
c_1 c_2 = (z_2 - z_1)^2 = (3 - i)^2
\]

### Step 5: Compute $ (3 - i)^2 $
Now, we compute the square of $ 3 - i $:
\[
(3 - i)^2 = 9 - 6i + (-1) = 8 - 6i
\]

### Final Answer:
The product of the two possible complex numbers $ c_1 $ and $ c_2 $ is $ \boxed{8 - 6i} $.

---

Do you want any further details or have any questions? Here are some follow-up questions that might help you explore this topic further:

1. What is the geometric interpretation of rotating complex numbers in the plane?
2. Can you find the exact coordinates of $ c_1 $ and $ c_2 $?
3. How does multiplying two complex numbers affect their magnitudes and arguments?
4. Why does rotating by $ 60^\circ $ and $ -60^\circ $ give two possible vertices of the triangle?
5. Can this method be generalized to finding vertices of other regular polygons?

**Tip:** When rotating a point in the complex plane, using exponential form of complex numbers can greatly simplify the computations!

There exist two complex numbers $c$, say $c_1$ and $c_2$, so that $3 + 2i$, $6 + i$, and $c$ form the vertices of an equilateral triangle. Find the product $c_1 c_2$ in rectangular form.

Explore how to find the product of two complex numbers that form the third vertex of an equilateral triangle with given points 3 + 2i and 6 + i in the complex plane. Learn the rotation of complex numbers and key geometric properties.

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