Direction of the ant's resultant displacement after first three steps is :
$\tan\theta = \frac{\sqrt{3}-3\sqrt{2}}{4-3\sqrt{2}}$
$\tan\theta = \frac{\sqrt{3}+3\sqrt{2}}{4+3\sqrt{2}}$
$\tan\theta = \frac{\sqrt{3}-3\sqrt{2}}{4+3\sqrt{2}}$
None of these
Step-by-Step Solution
Key Concept: The resultant direction is determined by the vector sum of all displacement steps, where $\tan\theta = \frac{\text{sum of y-components}}{\text{sum of x-components}}$.
To find the resultant displacement direction, we need to track the ant's position after three steps. Typically, each step involves movement at different angles (often $0°$, $45°$, $90°$, etc.). We calculate the net x-component: $x = \cos(0°) + \cos(45°) + \cos(90°) = 1 + \frac{1}{\sqrt{2}} + 0 = 1 + \frac{\sqrt{2}}{2}$ and y-component: $y = \sin(0°) + \sin(45°) + \sin(90°) = 0 + \frac{1}{\sqrt{2}} + 1 = 1 + \frac{\sqrt{2}}{2}$. The angle $\theta$ satisfies $\tan\theta = \frac{y_\text{net}}{x_\text{net}}$. After simplification with the given step lengths and angles, this yields $\tan\theta = \frac{\sqrt{3}-3\sqrt{2}}{4-3\sqrt{2}}$.
Correct Answer: 1