If `2z_1-3z_2 + z_3=0`, then `z_1, z_2 and z_3` are represented by

Options

A. three vertices of a triangle
B. three collinear points
C. three vertices of a rhombus
D. none of these

Answer

To solve the problem, we start with the given equation:

\[ 2z_1 - 3z_2 + z_3 = 0 \]

### Step 1: Rearranging the Equation
We can rearrange the equation to isolate \( z_3 \):

\[ z_3 = -2z_1 + 3z_2 \]

### Step 2: Expressing \( z_2 \)
Now, we can express \( z_2 \) in terms of \( z_1 \) and \( z_3 \):

\[ 3z_2 = 2z_1 + z_3 \]

Dividing both sides by 3 gives us:

\[ z_2 = \frac{2z_1 + z_3}{3} \]

### Step 3: Interpreting the Result
The expression we obtained for \( z_2 \):

\[ z_2 = \frac{2z_1 + z_3}{3} \]

This resembles the section formula in coordinate geometry, which states that if a point \( B \) divides the line segment \( AC \) in the ratio \( m:n \), then:

\[ B = \frac{mC + nA}{m+n} \]

In our case, \( z_2 \) is a point that divides the line segment joining \( z_1 \) and \( z_3 \) in the ratio \( 2:1 \).

### Step 4: Conclusion
Since \( z_2 \) is a point that lies on the line segment between \( z_1 \) and \( z_3 \), it implies that all three points \( z_1, z_2, z_3 \) are collinear.

Thus, the answer is that \( z_1, z_2, z_3 \) are **collinear points**.

### Final Answer:
**Collinear Points**

---