Question

If for complex numbers $(z_1$ and $ z_2)$ arg $(z_1 - z_2)$ = 0, then $| z_1 - z_2|$ is equal to

• A. $|z_1| + |z_2|$
• B. $|z_1| - |z_2|$
• C. $|\,|z_1| - |z_2|\,|$
• D. $0$

Answer 1

The Correct Option is C

We have $\left|z_{1}-z_{2}\right|^{2}=\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}-2\left|z_{1}\right|\left|z_{2}\right| cos$ $\left(\theta_{1}-\theta_{2}\right)$ where $\theta_{1}=arg. z_{1}, \theta_{2}=arg. z_{2}$ Since $arg. \left|z_{1}-z_{2}\right|=0$ $\therefore \left|z_{1}-z_{2}\right|^{2}=\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}-2\left|z_{1}\right|\left|z_{2}\right|$ $=\left(\left|z_{1}\right|-\left|z_{2}\right|^{2}\right)$ $\therefore \left|z_{1}-z_{2}\right|=|\left|z_{1}\right|-\left|z_{2}\right||$