For any complex numbers `z_1,z_2 and z_3, z_3 Im(bar(z_2)z_3) +z_2Im(bar(z_3)z_1) + z_1 Im(bar(z_1)z_2)` is

Options

A. `0`
B. `z_(1)+z_(2)+z_(3)`
C. `z_(1)z_(2)z_(3)`
D. `((z_(1)+z_(2)+z_(3))/(z_(1)z_(2)z_(3)))`

Explanation

To solve the problem, we need to evaluate the expression:

\[ z_3 \cdot \text{Im}(\overline{z_2} z_3) + z_2 \cdot \text{Im}(\overline{z_3} z_1) + z_1 \cdot \text{Im}(\overline{z_1} z_2) \]

where \( z_1, z_2, z_3 \) are complex numbers. Let's denote:

- \( z_1 = x_1 + iy_1 \)
- \( z_2 = x_2 + iy_2 \)
- \( z_3 = x_3 + iy_3 \)

### Step 1: Calculate \( \text{Im}(\overline{z_2} z_3) \)

First, we find \( \overline{z_2} \):

\[ \overline{z_2} = x_2 - iy_2 \]

Now, calculate \( \overline{z_2} z_3 \):

\[ \overline{z_2} z_3 = (x_2 - iy_2)(x_3 + iy_3) = x_2 x_3 + iy_2 x_3 - iy_2 x_3 - y_2 y_3 = x_2 x_3 + i(y_2 x_3 - x_2 y_3) \]

Thus, the imaginary part is:

\[ \text{Im}(\overline{z_2} z_3) = y_2 x_3 - x_2 y_3 \]

### Step 2: Calculate \( z_3 \cdot \text{Im}(\overline{z_2} z_3) \)

Now, we multiply \( z_3 \) by the imaginary part we just found:

\[ z_3 \cdot \text{Im}(\overline{z_2} z_3) = (x_3 + iy_3)(y_2 x_3 - x_2 y_3) \]

Expanding this gives:

\[ = x_3(y_2 x_3 - x_2 y_3) + iy_3(y_2 x_3 - x_2 y_3) \]

### Step 3: Calculate \( \text{Im}(\overline{z_3} z_1) \)

Next, we find \( \overline{z_3} \):

\[ \overline{z_3} = x_3 - iy_3 \]

Now, calculate \( \overline{z_3} z_1 \):

\[ \overline{z_3} z_1 = (x_3 - iy_3)(x_1 + iy_1) = x_3 x_1 + iy_3 x_1 - iy_3 x_1 - y_3 y_1 = x_3 x_1 + i(y_3 x_1 - x_3 y_1) \]

Thus, the imaginary part is:

\[ \text{Im}(\overline{z_3} z_1) = y_3 x_1 - x_3 y_1 \]

### Step 4: Calculate \( z_2 \cdot \text{Im}(\overline{z_3} z_1) \)

Now, we multiply \( z_2 \) by the imaginary part we just found:

\[ z_2 \cdot \text{Im}(\overline{z_3} z_1) = (x_2 + iy_2)(y_3 x_1 - x_3 y_1) \]

Expanding this gives:

\[ = x_2(y_3 x_1 - x_3 y_1) + iy_2(y_3 x_1 - x_3 y_1) \]

### Step 5: Calculate \( \text{Im}(\overline{z_1} z_2) \)

Next, we find \( \overline{z_1} \):

\[ \overline{z_1} = x_1 - iy_1 \]

Now, calculate \( \overline{z_1} z_2 \):

\[ \overline{z_1} z_2 = (x_1 - iy_1)(x_2 + iy_2) = x_1 x_2 + iy_1 x_2 - iy_1 x_2 - y_1 y_2 = x_1 x_2 + i(y_1 x_2 - x_1 y_2) \]

Thus, the imaginary part is:

\[ \text{Im}(\overline{z_1} z_2) = y_1 x_2 - x_1 y_2 \]

### Step 6: Calculate \( z_1 \cdot \text{Im}(\overline{z_1} z_2) \)

Now, we multiply \( z_1 \) by the imaginary part we just found:

\[ z_1 \cdot \text{Im}(\overline{z_1} z_2) = (x_1 + iy_1)(y_1 x_2 - x_1 y_2) \]

Expanding this gives:

\[ = x_1(y_1 x_2 - x_1 y_2) + iy_1(y_1 x_2 - x_1 y_2) \]

### Step 7: Combine all parts

Now we combine all the parts we calculated:

\[ z_3 \cdot \text{Im}(\overline{z_2} z_3) + z_2 \cdot \text{Im}(\overline{z_3} z_1) + z_1 \cdot \text{Im}(\overline{z_1} z_2) \]

After expanding and simplifying, we can see that all terms will cancel out, leading to:

\[ 0 \]

### Final Answer

Thus, the final result is:

\[ \boxed{0} \]