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Step 1: Identify the given complex number w
------------------------------------------
We are given that
$w = 1 - \sqrt{3}\,i.$
Step 2: Compute the modulus of w
--------------------------------
The modulus of a complex number $x + yi$ is given by $\sqrt{x^2 + y^2}$. Therefore,
$
|w|
= \sqrt{(1)^2 + \bigl(-\sqrt{3}\bigr)^2}
= \sqrt{1 + 3}
= \sqrt{4}
= 2.
$
Step 3: Relate |z| and |w| using |zw| = 1
-----------------------------------------
We are also given that
$
|zw| = 1.
$
Because $|zw| = |z| \cdot |w|$, we have
$
|z| \cdot |w| = 1.
$
Since $|w|=2$, this implies
$
|z| = \frac{1}{2}.
$
Step 4: Use the argument condition
----------------------------------
It is given that
$
\arg(z) - \arg(w) = \frac{\pi}{2}.
$
This means the angle between the vectors (complex numbers) $z$ and $w$ is $\frac{\pi}{2}$ radians, or $90^\circ$. In the triangle formed by the points $O$ (origin), $z$, and $w$, the angle at $O$ is a right angle.
Step 5: Interpret the triangle dimensions
-----------------------------------------
• One side (OW) has length $|w| = 2.$
• Another side (OZ) has length $|z| = \frac{1}{2}.$
• The angle $\angle ZOW$ is $\frac{\pi}{2}$ (right angle).
Hence, $\triangle ZOW$ is a right triangle with legs of lengths $2$ and $\tfrac{1}{2}$.
Step 6: Find the area of the triangle
-------------------------------------
For a right triangle with legs $a$ and $b$, the area is $\tfrac{1}{2} \times a \times b$. Here,
$
\text{Area} = \frac{1}{2} \times 2 \times \frac{1}{2} = \frac{1}{2}.
$
Therefore, the area of the triangle with vertices at the origin, $z$, and $w$ is
$
\boxed{\frac{1}{2}}.
$
