Condition for Points in Complex Plane to form Parallelogram/Mistake

Source Work

 * Chapter $1$: Complex Numbers
 * Supplementary Problems: $65$

This mistake can be seen in the 1981 printing of the second edition (1974) as published by Schaum: ISBN 0-070-84382-1

Mistake

 * Let $z_1, z_2, z_3, z_4$ be the position vectors of the vertices for quadrilateral $ABCD$. Prove that $ABCD$ is a parallelogram iff $z_1 - z_2 - z_3 + z_4 = 0$.

The unspoken assumption here is that $z_1 = A, z_2 = B, z_3 = C$ and $z_4 = D$. Without such an assumption, the question is ambiguous.

When defining a polygon in this manner, its vertices are cited in order around the perimeter of the polygon.

Thus the object defined should look like this:


 * ParallelogramInComplexPlane.png

By Condition for Points in Complex Plane to form Parallelogram it is seen that the correct condition is:


 * $z_1 - z_2 + z_3 - z_4 = 0$

Were the quadrilateral specified as $ABDC$, then:

Also see

 * Condition for Points in Complex Plane to form Parallelogram