© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Determine the magnitude of $w$
Given $w = 1 - \sqrt{3}\,i$. The magnitude of $w$ is
$|w| = \sqrt{(1)^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = 2$.
Step 2: Use the condition $|zw| = 1$ to find $|z|$
Since $|zw| = 1$, we get $|z|\cdot |w| = 1$. Therefore,
$|z| = \frac{1}{|w|} = \frac{1}{2}$.
Step 3: Use the argument condition $\arg(z) - \arg(w) = \frac{\pi}{2}$
This tells us that the complex number $z$ is obtained by rotating $w$ by
$\frac{\pi}{2}$ in the complex plane (i.e., $z$ is perpendicular to $w$)
and scaling its magnitude from 2 to $\frac{1}{2}$.
Step 4: Visualize or plot the points in the complex plane
We consider the vertices at the origin (0), $w$ (magnitude 2),
and $z$ (magnitude $\tfrac{1}{2}$, rotated
$\tfrac{\pi}{2}$ from $w$).
Step 5: Compute the area of the triangle
Since $z$ is perpendicular to $w$, the triangle with vertices at 0, $w$,
and $z$ forms a right triangle. Its legs have lengths $|w| = 2$
and $|z| = \tfrac{1}{2}$. Thus, the area is:
$$
\text{Area} = \frac{1}{2} \times |w| \times |z|
= \frac{1}{2} \times 2 \times \frac{1}{2}
= \frac{1}{2}.
$$
Final Answer
The area of the triangle (in square units) is $\frac{1}{2}$.
