Question

z1=-1,z2=i then find arg(z1/z2)​

Answer 1

Given :

• z1 = -1

• z2 = i

To find :

• arg(z1/z2)

Solution :

It is given that :- z1 = -1 and z2 = i

$\implies \sf \dfrac{z1}{z2} = \dfrac{ - 1}{i}$

[Where i = iota = √-1]

$\implies \sf \dfrac{z1}{z2} = \dfrac{ - 1}{i} \times \dfrac{i}{i}$

$\implies \sf \dfrac{z1}{z2} = \dfrac{ - i}{ ({i})^{2} }$

We know that :- i² = -1

$\implies \sf \dfrac{z1}{z2} = \dfrac{ - i}{ - 1}$

$\implies \sf \dfrac{z1}{z2} = \dfrac{ i}{1} = i$

Therefore, now :-

$\implies \sf arg \bigg( \dfrac{z1}{z2} \bigg)= arg \bigg( \dfrac{ - 1}{i} \bigg)$

$\implies \sf arg \bigg( \dfrac{ - 1}{i} \bigg) = arg( {i} )$

we know that i lies on positive( imaginary) axis.

$\implies \sf arg( {i} ) = \dfrac{\pi}{2}$

So we get :-

$\implies \sf arg \bigg( \dfrac{ - 1}{i} \bigg) = \dfrac{\pi}{2}$

Answer :

$\sf arg \bigg( \dfrac{z1}{z2} \bigg)= \dfrac{\pi}{2}$