Suppose w is a cube root of unity with `omega != 1`. Suppose P and Q are the points on the complex plane defined by `omega and omega^2`. If O is the origin, then what is the angle between OP and OQ?

Options

A. `60^(@)`
B. `90^(@)`
C. `120^(@)`
D. `150^(@)`

Answer

C

Explanation

` becauseomega=(-1)/(2)+i(sqrt3)/(2)`
`&omega^(2)=(-1)/(2)-i(sqrt3)/(2)`
`-(1)/(2)+(sqrt3)/(2)i" "(becauseomegane1)`

P and Q are points on complex plane. Angle between OP and OQ is
`theta=tan^(-1)|(m_(1)-m_(2))/(1+m_(1)m_(2))|`
`m_(1)"for line OP,"" "m_(2)"for line OQ"`
`m_(1)=((sqrt3)/(2)-0)/((-1)/(2)-0)" "m_(2)=((-sqrt3)/(2)-0)/((-1)/(2)-0)`
`impliesm_(1)=-sqrt3" "impliesm_(2)=sqrt3`
`thetatan^(-1)[(-sqrt3-sqrt3)/(1+(-sqrt3)(sqrt3))]`
`=tan^(-1)[(-2sqrt3)/(-2)]=pi-tan^(-1)sqrt3=pi-tan^(-1)tan""(pi)/(3)`
`theta=pi-(pi)/(3)`
`theta=120^(@)`