Maths · Modulus and argument (or amplitude) of a complex number
If are such that then is
If \( z_{1}, z_{2}, \varepsilon C \) are such that \( \left|z_{1}+z_{2}\right|^{2}= \) \( \left|z_{1}\right|^{2}+\left|z_{2}\right|^{2} \) then \( \frac{z_{1}}{z_{2}} \) is
- A. zero
- B. purely real
- C. purely imaginary
- D. complex
Step-by-step solution
For complex numbers, |z1+z2|^2 = |z1|^2 + |z2|^2 + 2Re(z1*conj(z2)). Given |z1+z2|^2 = |z1|^2 + |z2|^2, we get Re(z1*conj(z2))=0, so z1*conj(z2) is purely imaginary. Then z1/z2 = (z1*conj(z2))/|z2|^2, which is also purely imaginary (provided z2≠0). Hence, the ratio is purely imaginary.