Key Concepts and Formulas

• Complex Number Representation: A complex number $z = x + iy$ represents a point $(x, y)$ in the Argand plane.
• Translation as Complex Addition: Translating a point represented by $z_1$ by a displacement represented by $z_2$ results in a new position $z_1 + z_2$.
• Polar Form: A complex number can be represented in polar form as $z = r(\cos\theta + i\sin\theta)$, where $r$ is the magnitude and $\theta$ is the angle with respect to the positive real axis.

Step-by-Step Solution

Step 1: Initial Position

The point starts at $2 + i$.

• Why this step: This establishes the initial complex number representing the point's starting location in the Argand plane. $Z_0 = 2 + i$

Step 2: Movement 1 unit Eastwards

The point moves 1 unit eastwards.

• Why this step: Moving eastwards corresponds to adding a real number to the complex number. $Z_1 = Z_0 + 1$ $Z_1 = (2 + i) + 1$ $Z_1 = 3 + i$

Step 3: Movement 2 units Northwards

The point moves 2 units northwards.

• Why this step: Moving northwards corresponds to adding a positive imaginary number to the complex number. $Z_2 = Z_1 + 2i$ $Z_2 = (3 + i) + 2i$ $Z_2 = 3 + 3i$

Step 4: Movement $2\sqrt{2}$ units South-westwards

The point moves $2\sqrt{2}$ units in the south-west direction.

• Why this step: South-west corresponds to an angle of $225^\circ$ or $-135^\circ$ from the positive real axis. We need to find the complex number that represents this displacement and add it to the current position.
• Find the angle: The angle is $-135^\circ$.
• Calculate trigonometric values: $\cos(-135^\circ) = -\frac{\sqrt{2}}{2} = -\frac{1}{\sqrt{2}}$ $\sin(-135^\circ) = -\frac{\sqrt{2}}{2} = -\frac{1}{\sqrt{2}}$
• Construct the displacement complex number ($\Delta Z_3$): $\Delta Z_3 = 2\sqrt{2} \left( \cos(-135^\circ) + i\sin(-135^\circ) \right)$ $\Delta Z_3 = 2\sqrt{2} \left( -\frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}} \right)$ $\Delta Z_3 = -2 - 2i$
• Add the displacement to the current position: $Z_F = Z_2 + \Delta Z_3$ $Z_F = (3 + 3i) + (-2 - 2i)$ $Z_F = 1 + i$

Common Mistakes & Tips

• Direction and Angles: Always measure angles counter-clockwise from the positive real axis. Be especially careful with signs in different quadrants.
• Complex Addition: Make sure to add the real parts together and the imaginary parts together separately.

Summary

The problem involves a series of translations in the Argand plane, which can be easily solved by representing the point's position and each translation as complex numbers and then adding them. After moving 1 unit east, 2 units north, and $2\sqrt{2}$ units south-west, the final position of the point is $1 + i$.

The final answer is \boxed{1 + i}, which corresponds to option (B).