Let's break down the steps to determine the quadrant in which the difference [tex]\( z_7 - z_4 \)[/tex] is located.
### Step 1: Define the Complex Numbers
We start by defining the given complex numbers:
- [tex]\( z_1 = 5 \)[/tex] (This is a real number, equivalent to [tex]\( 5 + 0i \)[/tex])
- [tex]\( z_2 = -6i \)[/tex] (This is a purely imaginary number)
### Step 2: Calculate [tex]\( z_3 \)[/tex]
We are given that:
[tex]\[ z_3 = 3i + z_1 \][/tex]
Substitute [tex]\( z_1 \)[/tex] into the equation:
[tex]\[ z_3 = 3i + 5 \][/tex]
So, [tex]\( z_3 = 5 + 3i \)[/tex].
### Step 3: Calculate [tex]\( z_4 \)[/tex]
We are given that:
[tex]\[ z_4 = z_2 - 2 \][/tex]
Substitute [tex]\( z_2 \)[/tex] into the equation:
[tex]\[ z_4 = -6i - 2 \][/tex]
So, [tex]\( z_4 = -2 - 6i \)[/tex].
### Step 4: Calculate [tex]\( z_7 - z_4 \)[/tex]
To find [tex]\( z_7 - z_4 \)[/tex], we need to subtract [tex]\( z_4 \)[/tex] from [tex]\( z_3 \)[/tex]:
[tex]\[ z_7 - z_4 = z_3 - z_4 \][/tex]
Substitute the values of [tex]\( z_3 \)[/tex] and [tex]\( z_4 \)[/tex]:
[tex]\[ z_7 - z_4 = (5 + 3i) - (-2 - 6i) \][/tex]
Simplify the expression by distributing the negative sign and combining like terms:
[tex]\[ z_7 - z_4 = 5 + 3i + 2 + 6i \][/tex]
[tex]\[ z_7 - z_4 = (5 + 2) + (3i + 6i) \][/tex]
[tex]\[ z_7 - z_4 = 7 + 9i \][/tex]
### Step 5: Determine the Quadrant
The complex number [tex]\( 7 + 9i \)[/tex] has a real part [tex]\( 7 \)[/tex] and an imaginary part [tex]\( 9 \)[/tex].
- The real part [tex]\( 7 \)[/tex] is positive.
- The imaginary part [tex]\( 9 \)[/tex] is positive.
In the complex plane:
- When the real part is positive and the imaginary part is positive, the number is located in Quadrant I.
Therefore, [tex]\( z_7 - z_4 \)[/tex] is located in Quadrant I.
