In the Argand plane ,the vector $O P$ ,where $O$ is the origin and $P$ represents the complex number $z = 4 - 3 i$ ,is turned in the clockwise sense through $180^{\circ}$ and streched $3$ times. the complex number represented by the new vector is -----.

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Solution

$| z | = 5$ Let $z_{1}$ be the new vector obtained by rotating z in the clockwise sense through 180^., therefore $z_{1} = e^{- i π z} = (cos π - i sin π)$ ,i.e., z=−4+3i The unit vector in the direction of $z_{1}$ is $\to \frac{4}{5} + \frac{3}{5 i}$ . Therefore required vector $= 3 | z | (\frac{- 4}{5} + \frac{3}{5 i}) = 15 (\frac{- 4}{5} + \frac{3}{5 i}) = - 12 + 9 i$

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