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Question

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


A
eiπ3
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B
e2iπ3
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C
e4iπ3
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D
all the above
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Solution

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.$$\left\{ r=1 \right\} $$
$$ Z={ e }^{ i\theta  }\left\{ \theta ={ \pi  }/{ 3 } \right\} ={ 60 }^{ o  }$$
$$ Z={ e }^{ i \dfrac { \pi  }{ 3 }  }$$
Let $$\theta =\cfrac { \pi  }{ 3 } ,\cfrac { 2\pi  }{ 3 } ,\cfrac { 4\pi  }{ 3 } $$
 case$$(1)=\theta =\cfrac { \pi  }{ 3 } =a\left\{ Where\cfrac { \pi  }{ 3 } lies\quad in\quad { 1 }^{ st }\quad equation \right\}$$
 so argument$$ \theta =a$$
 Hence$$=Z={ e }^{ i\theta  }={ e }^{ i\dfrac { \pi  }{ 3 }  }$$
 Case$$(2)=a=\cfrac { 2\pi  }{ 3 } \left\{ Where\cfrac { 2\pi  }{ 3 } lies\quad in\quad { 2 }^{ nd }\quad equation \right\} $$
So argument$$ \theta =\pi -a$$
$$ =\pi -\cfrac { 2\pi  }{ 3 } $$
$$ =\cfrac { \pi  }{ 3 }$$
  Hence$$ z={ e }^{ i\dfrac { \pi  }{ 3 }  }$$
 Case$$(3)=a=\cfrac { 4\pi  }{ 3 } \left\{ Where\cfrac { 4\pi  }{ 3 } lies\quad in\quad { 3 }^{ rd }\quad equation \right\}$$
 So argument$$ \theta =\pi +a$$
$$=\pi +\cfrac { 4\pi  }{ 3 } $$
$$ =\cfrac { 7\pi  }{ 3 } =2\pi +\cfrac { \pi  }{ 3 } $$
 So argument $$ \tan\theta =\tan\left( 2\pi +\cfrac { \pi  }{ 3 }  \right) =\tan\cfrac { \pi  }{ 3 } $$
 Hence$$ z={ e }^{ i\dfrac { \pi  }{ 3 }  }$$
 Hence,unit vector can be represented as complex no.
$$ ={ e }^{ i\dfrac { \pi  }{ 3 }  },{ e }^{ i\dfrac { 2\pi  }{ 3 }  },{ e }^{ i\dfrac { 4\pi  }{ 3 }  }$$

886938_596399_ans_faab1479ec1d4684a4f23516903814d3.jpg

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