Area of the triangle formed by 3 complex numbers, `1+i,i-1,2i`, in the Argand plane, is

Options

A. `1//2`
B. 1
C. `sqrt(2)`
D. 2

Answer

To find the area of the triangle formed by the complex numbers \( z_1 = 1 + i \), \( z_2 = i - 1 \), and \( z_3 = 2i \) in the Argand plane, we can follow these steps:

### Step 1: Identify the complex numbers as points
We can represent the complex numbers as points in the Cartesian coordinate system:
- \( z_1 = 1 + i \) corresponds to the point \( (1, 1) \).
- \( z_2 = i - 1 \) corresponds to the point \( (-1, 1) \).
- \( z_3 = 2i \) corresponds to the point \( (0, 2) \).

### Step 2: Use the formula for the area of a triangle
The area \( A \) of a triangle formed by three points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

### Step 3: Substitute the coordinates into the formula
Substituting the coordinates of the points:
- \( (x_1, y_1) = (1, 1) \)
- \( (x_2, y_2) = (-1, 1) \)
- \( (x_3, y_3) = (0, 2) \)

The area becomes:
\[ A = \frac{1}{2} \left| 1(1 - 2) + (-1)(2 - 1) + 0(1 - 1) \right| \]

### Step 4: Simplify the expression
Calculating each term:
1. \( 1(1 - 2) = 1 \times (-1) = -1 \)
2. \( -1(2 - 1) = -1 \times 1 = -1 \)
3. \( 0(1 - 1) = 0 \)

Now, substituting these values back into the area formula:
\[ A = \frac{1}{2} \left| -1 - 1 + 0 \right| = \frac{1}{2} \left| -2 \right| = \frac{1}{2} \times 2 = 1 \]

### Final Answer
The area of the triangle formed by the complex numbers \( 1+i \), \( i-1 \), and \( 2i \) in the Argand plane is \( 1 \).

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