The problem you've provided involves three complex numbers $z_1$, $z_2$, and $z_3$, and we are given the following conditions:

$|z_1| = |z_2| = |z_3| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right| = 1$ We are asked to determine the value of $|z_1 + z_2 + z_3|$ using vectors.

Step-by-step approach:

1. Magnitude Condition: Each of the complex numbers has a modulus (magnitude) equal to 1, which means: $|z_1| = |z_2| = |z_3| = 1$ This tells us that each of these complex numbers lies on the unit circle in the complex plane.

2. Sum of Reciprocals: The equation also includes the condition: $\left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right| = 1$ Using the property of complex numbers, $|z^{-1}| = \frac{1}{|z|}$, and since $|z_1| = |z_2| = |z_3| = 1$, the magnitudes of the reciprocals are also 1. So: $\left| z_1^{-1} + z_2^{-1} + z_3^{-1} \right| = 1$

   By vector interpretation, the sum of the reciprocals behaves like a sum of vectors on the unit circle.

3. Geometric Interpretation: Since all complex numbers $z_1, z_2, z_3$ lie on the unit circle, they can be represented as points on the circle. The sum of their reciprocals suggests that these vectors are symmetric in some way, which leads us to conclude that $z_1, z_2, z_3$ might form an equilateral triangle on the unit circle.

4. Conclusion: For three complex numbers forming an equilateral triangle, the sum of the vectors $z_1 + z_2 + z_3$ will be zero. Therefore, the magnitude of this sum is: $|z_1 + z_2 + z_3| = 0$

Thus, the value of $|z_1 + z_2 + z_3|$ is 0.

Do you need more details on any part of this solution? Here are five related questions to deepen your understanding:

1. Why do the complex numbers lie on the unit circle based on the condition $|z_1| = |z_2| = |z_3| = 1$?
2. How does the property $|z^{-1}| = \frac{1}{|z|}$ affect the equation involving the reciprocals of $z_1$, $z_2$, and $z_3$?
3. What is the geometric interpretation of complex numbers forming an equilateral triangle on the unit circle?
4. How do symmetry and vectors help in solving this type of complex number problem?
5. Can you generalize this result for any other configuration of complex numbers on the unit circle?

Tip: Whenever working with complex numbers on the unit circle, think about the geometric properties of their arrangement (e.g., symmetry, angles) to simplify the calculations.