If `z_1 = (6,3), z_2 = (2,-1),` find `z_1/(z_2)`

Answer

To solve the problem of finding \( z_1 / z_2 \) where \( z_1 = (6, 3) \) and \( z_2 = (2, -1) \), we will follow these steps:

### Step 1: Convert the points to complex numbers
We represent the complex numbers from the given coordinates:
- \( z_1 = 6 + 3i \)
- \( z_2 = 2 - i \)

### Step 2: Write the division of the complex numbers
We need to calculate:
\[ \frac{z_1}{z_2} = \frac{6 + 3i}{2 - i} \]

### Step 3: Rationalize the denominator
To simplify the division, we multiply the numerator and the denominator by the conjugate of the denominator:
\[ \frac{6 + 3i}{2 - i} \cdot \frac{2 + i}{2 + i} \]

### Step 4: Multiply the numerators
Calculating the numerator:
\[ (6 + 3i)(2 + i) = 6 \cdot 2 + 6 \cdot i + 3i \cdot 2 + 3i \cdot i \]
\[ = 12 + 6i + 6i + 3i^2 \]
Since \( i^2 = -1 \), we have:
\[ = 12 + 12i + 3(-1) = 12 + 12i - 3 = 9 + 12i \]

### Step 5: Multiply the denominators
Calculating the denominator:
\[ (2 - i)(2 + i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5 \]

### Step 6: Combine the results
Now we can write the result of the division:
\[ \frac{9 + 12i}{5} \]
This simplifies to:
\[ \frac{9}{5} + \frac{12}{5}i \]

### Step 7: Final answer
Thus, the final result is:
\[ \frac{9}{5} + \frac{12}{5}i \]