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Step-by-Step Solution
Step 1: Understand the Problem
We are given four complex numbers that are vertices of a square in the Argand plane:
1. $z$
2. $\overline{z}$
3. $\overline{z} - 2\,\mathrm{Re}(\overline{z})$
4. $z - 2\,\mathrm{Re}(z)$
The side of this square is 4 units. We need to find the magnitude $\lvert z \rvert$.
Step 2: Connection Between $z$ and $\overline{z}$
Recall that for a complex number $z = x + i\,y$, its conjugate is $\overline{z} = x - i\,y$. Also,
$\mathrm{Re}(z)$ denotes the real part of $z$, which is $x$.
Hence:
- If $z = x + i\,y$, then $\overline{z} = x - i\,y$.
- $\mathrm{Re}(z) = x$ and $\mathrm{Re}(\overline{z}) = x$.
Step 3: Express the Given Vertices in Terms of $x$ and $y$
Let $z = x + i\,y$. Then:
1. $z = x + i\,y$.
2. $\overline{z} = x - i\,y$.
3. $\overline{z} - 2\,\mathrm{Re}(\overline{z}) = (x - i\,y) - 2x = -x - i\,y$.
4. $z - 2\,\mathrm{Re}(z) = (x + i\,y) - 2x = -x + i\,y$.
Step 4: Identify the Vertices Geometrically
In the Argand plane, these points can be written as:
1. $A: (x,\,y)$
2. $B: (x,\,-y)$
3. $C: (-x,\,-y)$
4. $D: (-x,\,y)$
Plotting these points suggests that $A$ and $C$ are opposite corners, and $B$ and $D$ are opposite corners, forming a square.
Step 5: Verify Side Length
Let us consider side $AB$ and ensure its length is 4.
- $A = (x,\,y)$
- $B = (x,\,-y)$
The length $AB = \sqrt{(x - x)^2 + (y - (-y))^2} = \sqrt{0 + (2y)^2} = \sqrt{4y^2} = 2\lvert y \rvert$.
Similarly, side $AD$ is:
- $A = (x,\,y)$
- $D = (-x,\,y)$
Length $AD = \sqrt{(x - (-x))^2 + (y - y)^2} = \sqrt{(2x)^2 + 0} = 2\lvert x \rvert$.
Since $AB$ and $AD$ are sides of the same square, their lengths should be equal:
$2\lvert y \rvert = 2\lvert x \rvert$
$\lvert y \rvert = \lvert x \rvert.$
We set $|x| = |y|$. Let us assume $x = y$ in magnitude (considering signs might just change orientation but won’t change the magnitude conclusion). Then each side must be 4:
$AB = 2|y| = 4 \quad \Rightarrow \quad |y| = 2.$
Hence $|x| = 2$ as well.
Step 6: Calculate $\lvert z\rvert$
If $z = x + i\,y$ and $|x|=|y|=2$, then:
$\lvert z \rvert = \sqrt{x^2 + y^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}.$
Step 7: Final Answer
Therefore, $\lvert z\rvert = 2\sqrt{2}.$
The correct answer matches Option 4: $2\sqrt{2}$.
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