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Question Number 140113 by EnterUsername last updated on 04/May/21

Let z_1 =1+i, z_2 =−1−i and z_3  be complex numbers  such that z_1 , z_2  and z_3  form an equilateral triangle.  Then z_3  is equal to  (A) (√3)(1+i)                                (B) (√3)(1−i)  (C) (√3)(i−1)                                (D) (√3)(−1−i)

$$\mathrm{Let}\:{z}_{\mathrm{1}} =\mathrm{1}+{i},\:{z}_{\mathrm{2}} =−\mathrm{1}−{i}\:\mathrm{and}\:{z}_{\mathrm{3}} \:\mathrm{be}\:\mathrm{complex}\:\mathrm{numbers} \\ $$$$\mathrm{such}\:\mathrm{that}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} \:\mathrm{and}\:{z}_{\mathrm{3}} \:\mathrm{form}\:\mathrm{an}\:\mathrm{equilateral}\:\mathrm{triangle}. \\ $$$$\mathrm{Then}\:{z}_{\mathrm{3}} \:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left(\mathrm{A}\right)\:\sqrt{\mathrm{3}}\left(\mathrm{1}+{i}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\sqrt{\mathrm{3}}\left(\mathrm{1}−{i}\right) \\ $$$$\left(\mathrm{C}\right)\:\sqrt{\mathrm{3}}\left({i}−\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:\sqrt{\mathrm{3}}\left(−\mathrm{1}−{i}\right) \\ $$

Commented by mr W last updated on 04/May/21

z_3  must be in 2^(nd)  or 4^(th)  quadrant,  therefore only (B) is correct.

$${z}_{\mathrm{3}} \:{must}\:{be}\:{in}\:\mathrm{2}^{{nd}} \:{or}\:\mathrm{4}^{{th}} \:{quadrant}, \\ $$$${therefore}\:{only}\:\left({B}\right)\:{is}\:{correct}. \\ $$

Commented by EnterUsername last updated on 04/May/21

Sir (C) has also been noted as correct answer.  How did you get there, please ?

$$\mathrm{Sir}\:\left(\mathrm{C}\right)\:\mathrm{has}\:\mathrm{also}\:\mathrm{been}\:\mathrm{noted}\:\mathrm{as}\:\mathrm{correct}\:\mathrm{answer}. \\ $$$$\mathrm{How}\:\mathrm{did}\:\mathrm{you}\:\mathrm{get}\:\mathrm{there},\:\mathrm{please}\:? \\ $$

Commented by mr W last updated on 04/May/21

i misread. (C) is in 2^(nd)  quadrant, so  it is also correct.

$${i}\:{misread}.\:\left({C}\right)\:{is}\:{in}\:\mathrm{2}^{{nd}} \:{quadrant},\:{so} \\ $$$${it}\:{is}\:{also}\:{correct}. \\ $$

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