Complex Numbers And Quadratic Equations question 318
Question: In the Argand plane, the vector $ z=4-3i $ is turned in the clockwise sense through $ 180^{o} $ and stretched three times. The complex number represented by the new vector is [DCE 2005]
Options:
A) $ 12+9i $
B) $ 12-9i $
C) $ -12-9i $
D) $ -12+9i $
Show Answer
Answer:
Correct Answer: D
Solution:
$ |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\pi }}z=(\cos \pi -i\sin \pi ), $ i.e., $ z=-4+3i $ 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+9i $
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