1. ## complex numbers

$\displaystyle P (Z_1)$ and $\displaystyle Q (Z_2)$ are complex numbers in argand diagram and $\displaystyle O$ is the origin.
show $\displaystyle OP$ and $\displaystyle OQ$ are perpendicular if

$\displaystyle |Z_1 - Z_2| = |Z_1 + Z_2|$

2. Hello, stud_02!

$\displaystyle P (Z_1)$ and $\displaystyle Q (Z_2)$ are complex numbers in argand diagram and $\displaystyle O$ is the origin.

Show $\displaystyle OP$ and $\displaystyle OQ$ are perpendicular if: .$\displaystyle |Z_1 - Z_2| = |Z_1 + Z_2|$
Code:
          Q * - - - - - - - - - - - * R
/  *                 *  /
/     *           *     /
/        *     *        /
Z2 /           *           /
/        *     *        /
/     *           *     /
/  *                 *  /
O * - - - - - - - - - - - * P
Z1

$\displaystyle \overrightarrow{Z_1} \:=\:\overrightarrow{OP}\:\text{ and }\:\overrightarrow{Z_2} \:=\:\overrightarrow{OQ}$

Then: .$\displaystyle \begin{array}{ccc}\overrightarrow{Z_1}+\overrighta rrow{Z_2} &=& \overrightarrow{OR} \\ \\[-4mm] \overrightarrow{Z_1}-\overrightarrow{Z_2} &=& \overrightarrow{QP} \end{array}$

Hence .$\displaystyle \begin{array}{ccc}|\overrightarrow{Z_1} + \overrightarrow{Z_2}|&=& OR \\ \\[-4mm] |\overrightarrow{Z_1}-\overrightarrow{Z_1}| &=& QP \end{array}$

If $\displaystyle |\overrightarrow{Z_1} - \overrightarrow{Z_2}| \:=\:|\overrightarrow{Z_1}+\overrightarrow{Z_2}|$, then: .$\displaystyle QP \:=\:OR$

. . And we have a parallelogram with equal diagonals.

A parallelogram with equal diagonals is a rectangle.

. . Therefore: .$\displaystyle OP \perp OQ$

3. Hello again, stud_02!

An algebraic proof is probably expected . . .

$\displaystyle P (Z_1)$ and $\displaystyle Q (Z_2)$ are complex numbers in the argand diagram and $\displaystyle O$ is the origin.

Show $\displaystyle OP \perp OQ$ if $\displaystyle |Z_1-Z_2| \:=\: |Z_1+Z_2|$
Let: .$\displaystyle \begin{array}{ccc}Z_1 &=& a + bi \\ Z_2 &=& c + di\end{array}$

We know that: .$\displaystyle Z_1 \perp Z_2\:\text{ if }\:Z_1\!\cdot\! Z_2 \:=\: 0$

. . That is: .$\displaystyle (a+bi)\cdot(c+di) \:=\:0 \quad\Rightarrow\quad ac + bd \:=\:0$ .[1]

We will show that [1] is true.

$\displaystyle Z_1 - Z_2 \;=\;(a-c) + (b-d)i$

. . $\displaystyle |Z_1-Z_2| \;=\;\sqrt{(a-c)^2 + (b-d)^2}$

$\displaystyle Z_1+Z_2 \;=\;(a+c)+(b+d)i$

. . $\displaystyle |Z_1+Z_2| \;=\;\sqrt{(a+c)^2 + (b+d)i}$

Since $\displaystyle |Z_1-Z_2| \:=\:|Z_1+Z_2|$, we have: .$\displaystyle \sqrt{(a-c)^2 + (b-d)^2} \;=\;\sqrt{(a+c)^2+(b+d)^2}$

Square both sides: .$\displaystyle (a-c)^2 + (b-d)^2 \;=\;(a+c)^2 + (b+d)^2$

Expand: .$\displaystyle a^2-2ac + c^2 + b^2 - 2bd + d^2 \;=\;a^2 + 2ac + c^2 + b^2 + 2bd + d^2$

. . which simplifies to: .$\displaystyle 4ac + 4bd \:=\:0 \quad\Rightarrow\quad\boxed{ ac + bd \:=\:0}$ . Q.E.D.

4. Originally Posted by stud_02
$\displaystyle P (Z_1)$ and $\displaystyle Q (Z_2)$ are complex numbers in argand diagram and $\displaystyle O$ is the origin.
show $\displaystyle OP$ and $\displaystyle OQ$ are perpendicular if

$\displaystyle |Z_1 - Z_2| = |Z_1 + Z_2|$

$\displaystyle |z_{1}-z_{2}|^2=|z_{1}+z_{2}|^2$

$\displaystyle |z_{1}|^2+|z_{2}|^2-z_{1} \bar z_{2}-\bar z_{1} z_{2}= |z_{1}|^2+|z_{2}|^2+z_{1}\bar z_{2}+\bar z_{1} z_{2}$

$\displaystyle 2(z_{1}\bar z_{2}+\bar z_{1} z_{2})=0$

$\displaystyle z_{1}\bar z_{2}=-\bar z_{1} z_{2}$

$\displaystyle \frac{z_{1}}{z_{2}}+\frac{\bar z_{1}}{\bar z_{2}}=0$

$\displaystyle \frac{z_{1}}{z_{2}}$ is purely imaginary

$\displaystyle arg(\frac{z_{1}}{z_{2}})=\frac{\pi}{2}$

$\displaystyle arg(z_{1})-arg(z_{2})=\frac{\pi}{2}$

$\displaystyle arg(\overrightarrow{OP})-arg(\overrightarrow{OQ})=\frac{\pi}{2}$

Hence,$\displaystyle \overrightarrow{OP}$ is perpendicular to $\displaystyle \overrightarrow{OQ}$