The points represented by the complex numbers `1+ i, -2+3i` and `( 5)/( 3) i` on the argand diagram are

Options

A. Vertices of an equilateral triangle
B. Vertices of an isosceles triangle
C. Collinear
D. None of the above

Answer

To determine the relationship between the points represented by the complex numbers \(1 + i\), \(-2 + 3i\), and \(\frac{5}{3}i\) on the Argand diagram, we can use the concept of the determinant to check if the points are collinear.

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( z_1 = 1 + i \) which corresponds to the point \( (1, 1) \).
- Let \( z_2 = -2 + 3i \) which corresponds to the point \( (-2, 3) \).
- Let \( z_3 = \frac{5}{3}i \) which corresponds to the point \( (0, \frac{5}{3}) \).

2. **Set Up the Determinant**:
To check if the points are collinear, we can set up the following determinant:
\[ \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \]
where \( (x_1, y_1) = (1, 1) \), \( (x_2, y_2) = (-2, 3) \), and \( (x_3, y_3) = (0, \frac{5}{3}) \).

The determinant becomes:
\[ \begin{vmatrix} 1 & 1 & 1 \\ -2 & 3 & 1 \\ 0 & \frac{5}{3} & 1 \end{vmatrix} \]

3. **Calculate the Determinant**:
We can calculate the determinant using the formula:
\[ D = x_1(y_2 - y_3) - y_1(x_2 - x_3) + 1(x_2y_3 - x_3y_2) \]
Plugging in the values:
\[ D = 1(3 - \frac{5}{3}) - 1(-2 - 0) + 1((-2)(\frac{5}{3}) - 0(3)) \]
Simplifying:
\[ D = 1\left(3 - \frac{5}{3}\right) + 2 - \frac{10}{3} \]
To simplify \(3 - \frac{5}{3}\), we convert 3 to a fraction:
\[ 3 = \frac{9}{3} \quad \Rightarrow \quad \frac{9}{3} - \frac{5}{3} = \frac{4}{3} \]
Thus:
\[ D = \frac{4}{3} + 2 - \frac{10}{3} \]
Converting 2 to a fraction:
\[ 2 = \frac{6}{3} \quad \Rightarrow \quad D = \frac{4}{3} + \frac{6}{3} - \frac{10}{3} \]
Combining:
\[ D = \frac{4 + 6 - 10}{3} = \frac{0}{3} = 0 \]

4. **Conclusion**:
Since the determinant \(D = 0\), the points are collinear.

### Final Answer:
The points represented by the complex numbers \(1 + i\), \(-2 + 3i\), and \(\frac{5}{3}i\) are **collinear**.

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