Question

The unit vector making an angle ${60}^o$ with the $x-$axis can be expressed as complex number by

• A. $e^{\frac{i\pi }{3}}$
• B. $e^{\frac{2i\pi }{3}}$
• C. $e^{\frac{4i\pi }{3}}$
• D. all the above

Answer 1

The correct option is D all the above

Given:-

A unit vector making an angle ${60}^o$with $x-axis.$

Solution:-

We have to represent the unit vector in complex .

Complex form

$Z=r{e}^{i\theta }$

Which is the modulus $\&\theta$ is argument.

So we can write as.$\{r=1\}$

$Z=e^{i\theta }\{\theta =\pi /3\}={60}^o$

$Z=e^{i\frac{\pi }{3}}$

Let $\theta =\frac{\pi }{3},\frac{2\pi }{3},\frac{4\pi }{3}$

case$\left(1\right)=\theta =\frac{\pi }{3}=a\left\{Where\frac{\pi }{3}liesin{1}^{st}equation\right\}$

so argument$\theta =a$

Hence$=Z=e^{i\theta }=e^{i\frac{\pi }{3}}$

Case$\left(2\right)=a=\frac{2\pi }{3}\left\{Where\frac{2\pi }{3}liesin{2}^{nd}equation\right\}$

So argument$\theta =\pi -a$

$=\pi -\frac{2\pi }{3}$

$=\frac{\pi }{3}$

Hence$z=e^{i\frac{\pi }{3}}$

Case$\left(3\right)=a=\frac{4\pi }{3}\left\{Where\frac{4\pi }{3}liesin{3}^{rd}equation\right\}$

So argument$\theta =\pi +a$

$=\pi +\frac{4\pi }{3}$

$=\frac{7\pi }{3}=2\pi +\frac{\pi }{3}$

So argument $\tan \theta =\tan (2\pi +\frac{\pi }{3})=\tan \frac{\pi }{3}$

Hence$z=e^{i\frac{\pi }{3}}$

Hence,unit vector can be represented as complex no.

$=e^{i\frac{\pi }{3}},e^{i\frac{2\pi }{3}},e^{i\frac{4\pi }{3}}$