Question

A particle P starts from $Z_0 = 1 + 2i$. It moves horizontally away from origin by 5 units, then vertically up by 3 units to $Z_1$. From $Z_1$ it moves $\sqrt{2}$ units in direction $\hat{i} + \hat{j}$, then moves through $\pi/2$ anticlockwise on a circle with centre at origin to reach $Z_2$. Then $Z_2 = \dots$

• A. $6 + 7i$
• B. -7 + 6i
• C. -6 + 7i
• D. 7 - 6i

Hint:

Argand plane geometric transformations are incredibly simple: Translation by $(a,b)$ is just adding $(a+bi)$. Rotation around the origin by an angle $\theta$ is just multiplying by $e^{i\theta} = (\cos\theta + i\sin\theta)$. For a pure $90^\circ$ turn, just multiply by $i$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This problem represents a series of physical translations and rotations applied sequentially to a point on the complex Argand plane.

Step 2: Detailed Explanation:
Track the position of the particle step-by-step:
1. Initial position: $Z_0 = 1 + 2i$.
2. Horizontal movement: It moves 5 units horizontally away from the origin (which means moving in the positive real direction, $+5$).
$$Z_{\text{temp1}} = (1 + 2i) + 5 = 6 + 2i$$
3. Vertical movement: It moves vertically up by 3 units (which means moving in the positive imaginary direction, $+3i$) to reach $Z_1$.
$$Z_1 = (6 + 2i) + 3i = 6 + 5i$$
4. Diagonal movement: It moves $\sqrt{2}$ units in the direction of the vector $\hat{i} + \hat{j}$.
The complex number corresponding to the direction vector $\hat{i} + \hat{j}$ is $1 + i$.
The magnitude of $1+i$ is $\sqrt{1^2+1^2} = \sqrt{2}$.
Since the required movement distance is exactly $\sqrt{2}$ units, we simply add the complex vector $1+i$ to the current position.
$$Z_{\text{temp2}} = Z_1 + (1 + i) = (6 + 5i) + (1 + i) = 7 + 6i$$
5. Rotation: Finally, it rotates by an angle of $\pi/2$ ($90^\circ$) anticlockwise about the origin to reach $Z_2$.
Multiplying a complex number by $i$ mathematically perfectly rotates it by $90^\circ$ anticlockwise about the origin.
$$Z_2 = (7 + 6i) \times i = 7i + 6i^2$$
Since $i^2 = -1$:
$$Z_2 = -6 + 7i$$

Step 4: Final Answer:
The final coordinate is $-6 + 7i$, matching option (c).