Condition for Points in Complex Plane to form Parallelogram/Mistake

Source Work

1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.)

Chapter $1$: Complex Numbers

Supplementary Problems: $65$

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 if and only if $z_1 - z_2 - z_3 + z_4 = 0$.

Correction

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:

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:

\(\ds \vec {AB}\)

\(=\)

\(\ds \vec {CD}\)

\(\ds \leadstoandfrom \ \ \)

\(\ds z_2 - z_1\)

\(=\)

\(\ds z_4 - z_3\)

\(\ds \leadstoandfrom \ \ \)

\(\ds z_1 - z_2 - z_3 + z_4\)

\(=\)

\(\ds 0\)

Also see

• Condition for Points in Complex Plane to form Parallelogram

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Graphical Representation of Complex Numbers. Vectors: $65$