In the Argand plane, the vector $z = 4 - 3 i$ is turned in the clockwise sense through 180 $^{o}$ and stretched three times. The complex number represented by the new vector is

12 + 9i

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12 - 9i

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-12 - 9i

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-12 + 9i

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Solution

The correct option is D -12 + 9i
$| z | = \sqrt{4^{2} + (- 3)^{2}} = 5$
Let z $_{1}$ be the new vector obtained by rotating z in the clockwise sense through 180 $^{o}$ ,
therefore $z_{1} = e^{- i π} z = (c o s π - i s i n π)$ . i.e. $z = - 4 + 3 i$
The unit vector in the direction of z $- 1$ is $- \frac{4}{5} + \frac{3}{5} i$
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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In the Argand plane, the vector z=4−3i is turned in the clockwise sense through 180o and stretched three times. The complex number represented by the new vector is

In the Argand plane, the vector $z = 4 - 3 i$ is turned in the clockwise sense through 180 $^{o}$ and stretched three times. The complex number represented by the new vector is