Rotating a complex Number

Algebra Level 3

If z = 3 − 4 ι \large z = 3 - 4\iota is turned π 2 \dfrac{\pi}{2} in anti - clockwise direction , then the new position of z z is

Details : ι = − 1 \iota = \sqrt{-1}

Original Problem
4 + 3 ι 4 + 3\iota None of these − 4 − 3 ι -4 - 3\iota − 3 − 4 ι -3 - 4\iota 3 + 4 ι 3 + 4\iota 3 − 4 ι 3 - 4\iota 4 − 3 ι 4 - 3\iota − 3 + 4 ι -3 + 4\iota

Akhil Bansal by Akhil Bansal , 22, India

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3 solutions

Chew-Seong Cheong
Nov 12, 2015

Turning a complex number z z by π 2 \dfrac{\pi}{2} in the complex plane (Argand diagram) is equivalent to multiplying z z with i i . ⇒ z i = ( 3 − 4 i ) i = 3 i − 4 i 2 = 4 + 3 i \Rightarrow zi = (3-4i)i = 3i-4i^2 = \boxed{4+3i} .

8 Helpful 0 Interesting 0 Brilliant 0 Confused
Sharath K Menon
Nov 13, 2015

Using Rotation theorem about origin

z ′ z \frac{z'}{z} = ∣ z ′ ∣ ∣ z ∣ \frac{|z'|}{|z|} * ( cos ⁡ θ + i sin ⁡ θ (\cos\theta+i\sin\theta )

where θ \theta = π \pi /2

the magnitudes are equal i.e |z'| = |z|

Thus, z' = iz Thus z= 4+3i

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Sarthak Singla
Jan 9, 2018

just draw and see.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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