question_answer

The points \[{{z}_{1}},{{z}_{2}},{{z}_{3}},{{z}_{4}}\] in a complex plane are vertices of a parallelogram taken in order, then

A) \[{{z}_{1}}+{{z}_{4}}={{z}_{2}}+{{z}_{3}}\]

B) \[{{z}_{1}}+{{z}_{3}}={{z}_{2}}+{{z}_{4}}\]

C) \[{{z}_{1}}+{{z}_{2}}={{z}_{3}}+{{z}_{4}}\]

D) None of these

Correct Answer: B

Solution :

Let \[{{z}_{1}},\,\,{{z}_{2}}\,\,{{z}_{3}}\,\,and\,\,{{z}_{4}}\] the points in complex lane be the vertices of a parallelogram taken in order. Since the diagonals of a parallelogram bisect, hence the mid points of AC and BD must coincide i.e., \[\frac{{{z}_{1}}+{{z}_{3}}}{2}=\frac{{{z}_{2}}+{{z}_{4}}}{2}\,\,\Rightarrow \,\,{{z}_{1}}+{{z}_{3}}={{z}_{2}}+{{z}_{4}}\]