The vertices A, B of a square ABCD are given to be `z_(1), z_(2)` then the vertices ` z_(3) , z_(4)` in terms of ` z_(1) , z_(2)` are . . . .

Answer

To find the vertices \( z_3 \) and \( z_4 \) of square ABCD in terms of \( z_1 \) and \( z_2 \), we can follow these steps:

### Step 1: Understand the Geometry of the Square
Given that \( z_1 \) and \( z_2 \) are the vertices A and B of the square, we know that the line segment AB is one side of the square. The angle between the sides of the square is \( 90^\circ \) or \( \frac{\pi}{2} \) radians.

### Step 2: Use the Rotation Property
The vertices \( z_3 \) and \( z_4 \) can be found by rotating the vector from \( z_1 \) to \( z_2 \) by \( 90^\circ \) counterclockwise to get \( z_3 \), and then again by \( 90^\circ \) counterclockwise to get \( z_4 \).

### Step 3: Find \( z_3 \)
Using the rotation formula, we can express \( z_3 \) as follows:
\[ z_3 = z_2 + i(z_2 - z_1) \]
This equation arises because rotating a point \( z_1 \) around \( z_2 \) by \( 90^\circ \) involves adding \( i(z_2 - z_1) \) to \( z_2 \).

### Step 4: Simplify the Expression for \( z_3 \)
Now, we can simplify the expression for \( z_3 \):
\[ z_3 = z_2 + i(z_2 - z_1) = z_2 + iz_2 - iz_1 = (1 + i)z_2 - iz_1 \]

### Step 5: Find \( z_4 \)
To find \( z_4 \), we can rotate \( z_2 \) to find \( z_3 \) and then rotate \( z_3 \) to find \( z_4 \):
\[ z_4 = z_3 + i(z_3 - z_2) \]
Substituting \( z_3 \) into this equation:
\[ z_4 = ((1 + i)z_2 - iz_1) + i(((1 + i)z_2 - iz_1) - z_2) \]
Simplifying this expression:
\[ z_4 = ((1 + i)z_2 - iz_1) + i((iz_2 - iz_1)) \]
\[ = ((1 + i)z_2 - iz_1) + (-z_2 + iz_1) = (1 + i - 1)z_2 + (i - i)z_1 \]
\[ = iz_2 + (1 - i)z_1 \]

### Final Expressions
Thus, the vertices \( z_3 \) and \( z_4 \) in terms of \( z_1 \) and \( z_2 \) are:
\[ z_3 = (1 + i)z_2 - iz_1 \]
\[ z_4 = iz_2 + (1 - i)z_1 \]