If |z₁+i(z₃-z₁)|=|z₁|+|z₃-z₁|.

Then how can we prove that,

arg{(z₁)/(z₃-z₁)}=pie/2.

To tackle the problem you've presented, we need to analyze the equation given and understand the implications of the absolute values involved. The equation states that the magnitude of a complex expression is equal to the sum of the magnitudes of its components. This can lead us to some interesting conclusions about the arguments of the complex numbers involved.

Breaking Down the Equation

We start with the equation:

|z₁ + i(z₃ - z₁)| = |z₁| + |z₃ - z₁|

Here, z₁ and z₃ are complex numbers. The term i(z₃ - z₁) represents a rotation of the vector (z₃ - z₁) by 90 degrees in the complex plane. The left side of the equation represents the magnitude of this rotated vector added to z₁.

Understanding the Geometric Interpretation

In the complex plane, the equality of magnitudes can be interpreted geometrically. The equation states that the length of the vector z₁ + i(z₃ - z₁) is equal to the sum of the lengths of z₁ and (z₃ - z₁). This can only happen if the vectors are aligned in a specific way.

• The left side represents a triangle formed by the points corresponding to z₁, z₃, and the origin.
• The right side suggests that the triangle inequality holds with equality, which occurs when the points are collinear.

Establishing the Argument Condition

For the equality to hold, the vectors must be in the same direction or exactly opposite. This leads us to conclude that the angle between z₁ and (z₃ - z₁) must be 90 degrees. In terms of arguments, this means:

arg(z₁) - arg(z₃ - z₁) = π/2

Rearranging this gives us:

arg(z₁) = arg(z₃ - z₁) + π/2

Final Steps to Prove the Argument Condition

From the above relationship, we can express the ratio of the two complex numbers:

arg{(z₁)/(z₃ - z₁)} = arg(z₁) - arg(z₃ - z₁)

Substituting our earlier result:

arg{(z₁)/(z₃ - z₁)} = (arg(z₃ - z₁) + π/2) - arg(z₃ - z₁)

This simplifies to:

arg{(z₁)/(z₃ - z₁)} = π/2

Conclusion

Thus, we have shown that if the original condition holds, it leads us to conclude that the argument of the ratio of z₁ to (z₃ - z₁) is indeed equal to π/2. This indicates that the vector z₁ is perpendicular to the vector (z₃ - z₁) in the complex plane, confirming the geometric interpretation of the problem.