Position of Particle After Translation and Rotation in Complex Plane

To find the final position of the particle $z_2$, we break down its movement step-by-step:

1. Initial Position ($z_0$):
   The particle starts at the point:
   $z_0 = 1 + 2i$
   In Cartesian coordinates, this corresponds to the point $(1, 2)$.

2. Movement to $z_1$:

   • Horizontal movement: Since the initial $x$-coordinate is $1$ (which is positive), moving away from the origin horizontally by $5$ units means moving in the positive $x$-direction:
     $x_1 = 1 + 5 = 6$
   • Vertical movement: Since the initial $y$-coordinate is $2$ (which is positive), moving away from the origin vertically by $3$ units means moving in the positive $y$-direction:
     $y_1 = 2 + 3 = 5$

   Therefore, the complex position $z_1$ is:
   $z_1 = 6 + 5i$

3. Movement along the vector $\hat{i} + \hat{j}$:

   • The unit vector in the direction of $\hat{i} + \hat{j}$ is:
     $\hat{u} = \frac{\hat{i} + \hat{j}}{|\hat{i} + \hat{j}|} = \frac{\hat{i} + \hat{j}}{\sqrt{1^2 + 1^2}} = \frac{\hat{i} + \hat{j}}{\sqrt{2}}$
   • Moving a distance of $\sqrt{2}$ units in this direction gives a displacement vector:
     $\vec{d} = \sqrt{2} \left(\frac{\hat{i} + \hat{j}}{\sqrt{2}}\right) = \hat{i} + \hat{j}$
   • In complex number form, this displacement is $1 + i$.
   • Let $z_1'$ be the new position after this translation:
     $z_1' = z_1 + (1 + i) = (6 + 5i) + (1 + i) = 7 + 6i$

4. Rotation to reach $z_2$:

   • Moving through an angle of $\frac{\pi}{2}$ in the anticlockwise direction along a circle centered at the origin corresponds to multiplying the complex number $z_1'$ by $e^{i\frac{\pi}{2}} = i$.
   • Therefore, the final point $z_2$ is:
     $z_2 = z_1' \cdot e^{i\frac{\pi}{2}} = (7 + 6i) \cdot i = 7i + 6i^2 = -6 + 7i$

Thus, the point $z_2$ is given by $-6 + 7i$.

Correct Option: D