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If the complex number Z(1) and Z(2), arg...

If the complex number `Z_(1)` and `Z_(2), arg (Z_(1))- arg(Z_(2)) =0`. then show that `|z_(1)-z_(2)| = |z_(1)|-|z_(2)|`.

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To solve the problem, we need to show that if the complex numbers \( Z_1 \) and \( Z_2 \) satisfy the condition \( \arg(Z_1) - \arg(Z_2) = 0 \), then it follows that \( |Z_1 - Z_2| = |Z_1| - |Z_2| \). ### Step-by-Step Solution: 1. **Understanding the Argument Condition**: Since \( \arg(Z_1) - \arg(Z_2) = 0 \), we can conclude that \( \arg(Z_1) = \arg(Z_2) \). This means that both complex numbers \( Z_1 \) and \( Z_2 \) have the same angle (argument) in the polar coordinate system. 2. **Expressing \( Z_1 \) and \( Z_2 \) in Polar Form**: ...
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