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Thread: complex number

  1. #1
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    complex number

    A and B are the points representing the complex numbers z and (1 + i )z in an Argand diagram. Prove that triangle OAB is a right-angled isoceles triangle, where O is the origin.
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    MHF Contributor FernandoRevilla's Avatar
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    Re: complex number

    Hints: $\displaystyle |OA|=|z|,\;|OB|=\sqrt{2}|z|$ . On the other hand $\displaystyle |AB|=|(1+i)z-z|=|iz|=|z|$ and $\displaystyle 1+i=\sqrt{2}e^{\pi i/4}$ .
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  3. #3
    Member kalyanram's Avatar
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    Re: complex number

    Given:
    $\displaystyle OA = z, OB = (1 + i)z $
    By the definition of multiplication of complex numbers if $\displaystyle z_1 = r_1. cis {\theta}_1, z_1 = r_2. cis {\theta}_2$ we have
    $\displaystyle z_1.z_2 = r_1.r_2 cis({\theta}_1 + {\theta}_2)$ so we have $\displaystyle z_1 = (1 + i) = \sqrt 2 .cis {\frac{\pi}{4}}$.
    $\displaystyle \therefore$ in $\displaystyle \Delta OAB, |OB| = \sqrt 2.|OA|, \; \angle AOB = \frac{\pi}{4}$

    $\displaystyle \therefore \Delta OAB $ is right isosceles triangle.

    Kalyan
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