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  4. If z1,z2 and z3 are any three complex numbers, then the fourth vertex of the parallelogram whose three vertices are z1,z2 and z3 taken in order, is:

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Q. If $ {{z}_{1}},{{z}_{2}} $ and $ {{z}_{3}} $ are any three complex numbers, then the fourth vertex of the parallelogram whose three vertices are $ {{z}_{1}},{{z}_{2}} $ and $ {{z}_{3}} $ taken in order, is:

KEAMKEAM 2003

Solution:

Let z be the fourth vertex of parallelogram, then $ \frac{{{z}_{1}}+{{z}_{3}}}{2}=\frac{{{z}_{2}}+z}{2} $ $ \Rightarrow $ $ z={{z}_{1}}+{{z}_{3}}-{{z}_{2}} $