Final answer:
The value of \(z_2\) can be determined by adding point C, \(z_3\), to the 90-degree rotated vector from C to A, giving the answer as \(z_2 = z_3 + i(z_1 - z_3)\).
Step-by-step explanation:
The question involves complex numbers and geometry, specifically relate to a triangle in the complex plane. Given two conditions: first that the angle ABC is \(\frac{\pi}{4}\) radians (or 45 degrees), and second that the ratio of the lengths AB to BC is \(\sqrt{2}\), we are asked to find the value of \(z_2\), the complex number representing point B.
The given conditions imply that triangle ABC is an isosceles right triangle (a 45-45-90 triangle). Therefore, the length from A to B is \(\sqrt{2}\) times the length from B to C, and since the angle at B is 45 degrees, the sides opposite and adjacent to this angle are congruent. By using rotations in the complex plane, we know that multiplying by \(i\) represents a 90-degree counterclockwise rotation.
To find \(z_2\), we need to add to \(z_3\) a vector that points from C to B. Since AB is \(\sqrt{2}\) times BC and AB is at a 45-degree rotation relative to BC, this means that the complex number representing the side AB (from point A to point B) can be obtained by taking the complex number representing BC (from point B to point C), which is \(z_1 - z_3\), then rotating it by 45 degrees which is equivalent to multiplying by \(\frac{1 + i}{\sqrt{2}}\) to maintain the length proportion.
The actual length adjustment is not needed since the options do not include scalar multiplication, only a rotation can be deduced to apply.
Thus, the correct option is to add point C, \(z_3\), to the rotated vector \(i(z_1 - z_3)\). Therefore, the correct answer is option C: \(z_3 + i(z_1 - z_3)\).