Step-by-Step Calculation of the Particle's Position

We can find the final point $z_3$ by tracking the particle's position in the complex plane after each described movement. We can treat the complex number $x + iy$ as a point $(x, y)$.

1. Initial Position ($z_0$)

The particle starts at $z_0 = 1 + 2i$, which corresponds to the point P_0 = (1, 2).

2. Finding Position $z_1$

• Horizontal movement: The particle moves 5 units horizontally "away from the origin". Since the point (1, 2) is in the first quadrant (positive x), moving away from the origin means moving in the positive x-direction.
  New x-coordinate = 1 + 5 = 6. The point is now (6, 2).
• Vertical movement: It then moves 3 units vertically "away from the origin". Since the point (6, 2) is in the first quadrant (positive y), this means moving in the positive y-direction.
  New y-coordinate = 2 + 3 = 5. The point is now (6, 5).

Therefore, $z_1$ corresponds to the point (6, 5), so $z_1 = 6 + 5i$.

3. Finding Position $z_2$

From $z_1$, the particle moves $\sqrt{2}$ units in the direction of the vector $\hat{i} + \hat{j}$.

• The direction $\hat{i} + \hat{j}$ corresponds to a vector (1, 1).
• A movement of $\sqrt{2}$ units in this direction results in a displacement of (1, 1).
• New position = Old position + Displacement = (6, 5) + (1, 1) = (7, 6).

Therefore, $z_2$ corresponds to the point (7, 6), so $z_2 = 7 + 6i$.

4. Finding Position $z_3$

From $z_2$, the particle moves through an angle of $\frac{\pi}{2}$ (or 90°) in an anti-clockwise direction around the origin.

• In the complex plane, rotating a number by $\frac{\pi}{2}$ anti-clockwise is equivalent to multiplying it by 'i'.
• $z_3 = z_2 \times i$
• $z_3 = (7 + 6i) \times i$
• $z_3 = 7i + 6i^2$
• Since $i^2 = -1$, this becomes: $z_3 = 7i + 6(-1) = -6 + 7i$.

Conclusion

The point $z_3$ is given by -6 + 7i.