question_answer

The complex numbers \[{{z}_{1}},{{z}_{2}},{{z}_{3}}\] are the vertices of a triangle. Then the complex numbers \[z\] which make the triangle into a parallelogram is

A) \[{{z}_{1}}+{{z}_{2}}-{{z}_{3}}\]

B) \[{{z}_{1}}-{{z}_{2}}+{{z}_{3}}\]

C) \[{{z}_{2}}+{{z}_{3}}-{{z}_{1}}\]

D) All the above

Correct Answer: D

Solution :

Let \[A,B,C\] be the points represented by the numbers \[{{z}_{1}},{{z}_{2}},{{z}_{3}}\]and P be the point represented by \[z\] Now the four points \[A,B,C,P\] form a parallelogram in the following three orders.   (i) \[A,B,P,C\](ii) \[B,C,P,A\]and (iii) \[C,A,P,B\] In case (i), the condition for \[A,B,P,C\]to form a parallelogram is  \[\overrightarrow{AB}=\overrightarrow{CP}\] i.e., \[{{z}_{2}}-{{z}_{1}}=z-{{z}_{3}}\] or \[z={{z}_{2}}+{{z}_{3}}-{{z}_{1}}\] Similarly in case (ii) and (iii), the required points \[\overrightarrow{BC}=\overrightarrow{AP}\]or \[{{z}_{3}}-{{z}_{2}}=z-{{z}_{1}}\]i.e., \[z={{z}_{3}}+{{z}_{1}}-{{z}_{2}}\] and  \[\overrightarrow{CA}=\overrightarrow{BP}\]or \[{{z}_{1}}-{{z}_{3}}=z-{{z}_{2}}\]i.e., \[z={{z}_{1}}+{{z}_{2}}-{{z}_{3}}\]