A bird starts from `(1,0,0)` in the direction `(2hati+3hatj-6hatk)` with a speed `21 m//s` for `5` sec,then along the direction `(3hati+4hatj+5hatk)`with a speed `5sqrt2m//s` for `5` sec. Find the final displacement of the bird so that it reaches the origin?

Options

A. `30hati+45hatj-45hatk`
B. `-45hati-65hatj+65hatk`
C. `-46hati-65hatj+65hatk`
D. `46hati-65hatj-65hatk`

Explanation

To solve the problem step by step, we need to calculate the displacement of the bird during its flight and then find the final displacement required to reach the origin.

### Step 1: Calculate the displacement during the first segment of the flight.
The bird starts from the point \( (1, 0, 0) \) and flies in the direction of \( (2\hat{i} + 3\hat{j} - 6\hat{k}) \) with a speed of \( 21 \, \text{m/s} \) for \( 5 \, \text{s} \).

1. **Calculate the unit vector in the direction of the first segment:**
\[ \text{Magnitude} = \sqrt{2^2 + 3^2 + (-6)^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]
\[ \text{Unit vector} = \frac{(2\hat{i} + 3\hat{j} - 6\hat{k})}{7} = \frac{2}{7}\hat{i} + \frac{3}{7}\hat{j} - \frac{6}{7}\hat{k} \]

2. **Calculate the displacement vector \( S_1 \):**
\[ S_1 = \text{Speed} \times \text{Time} \times \text{Unit vector} = 21 \, \text{m/s} \times 5 \, \text{s} \times \left(\frac{2}{7}\hat{i} + \frac{3}{7}\hat{j} - \frac{6}{7}\hat{k}\right) \]
\[ S_1 = 105 \left(\frac{2}{7}\hat{i} + \frac{3}{7}\hat{j} - \frac{6}{7}\hat{k}\right) = 30\hat{i} + 45\hat{j} - 90\hat{k} \]

### Step 2: Calculate the displacement during the second segment of the flight.
The bird then flies in the direction of \( (3\hat{i} + 4\hat{j} + 5\hat{k}) \) with a speed of \( 5\sqrt{2} \, \text{m/s} \) for \( 5 \, \text{s} \).

1. **Calculate the unit vector in the direction of the second segment:**
\[ \text{Magnitude} = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} = 5\sqrt{2} \]
\[ \text{Unit vector} = \frac{(3\hat{i} + 4\hat{j} + 5\hat{k})}{5\sqrt{2}} = \frac{3}{5\sqrt{2}}\hat{i} + \frac{4}{5\sqrt{2}}\hat{j} + \frac{5}{5\sqrt{2}}\hat{k} \]

2. **Calculate the displacement vector \( S_2 \):**
\[ S_2 = \text{Speed} \times \text{Time} \times \text{Unit vector} = 5\sqrt{2} \, \text{m/s} \times 5 \, \text{s} \times \left(\frac{3}{5\sqrt{2}}\hat{i} + \frac{4}{5\sqrt{2}}\hat{j} + \frac{5}{5\sqrt{2}}\hat{k}\right) \]
\[ S_2 = 25 \left(\frac{3}{5\sqrt{2}}\hat{i} + \frac{4}{5\sqrt{2}}\hat{j} + \frac{5}{5\sqrt{2}}\hat{k}\right) = 15\hat{i} + 20\hat{j} + 25\hat{k} \]

### Step 3: Calculate the total displacement from the initial position.
The total displacement \( S \) from the initial position \( (1, 0, 0) \) is given by:
\[ S = S_1 + S_2 = (30\hat{i} + 45\hat{j} - 90\hat{k}) + (15\hat{i} + 20\hat{j} + 25\hat{k}) \]
\[ S = (30 + 15)\hat{i} + (45 + 20)\hat{j} + (-90 + 25)\hat{k} = 45\hat{i} + 65\hat{j} - 65\hat{k} \]

### Step 4: Find the final displacement required to reach the origin.
To find the final displacement \( S_3 \) required to reach the origin, we set up the equation:
\[ \text{Final Position} = \text{Initial Position} + S + S_3 = (1, 0, 0) + (45\hat{i} + 65\hat{j} - 65\hat{k}) + S_3 = (0, 0, 0) \]
This leads to:
\[ S_3 = - (1 + 45)\hat{i} - 65\hat{j} + 65\hat{k} = -46\hat{i} - 65\hat{j} + 65\hat{k} \]

### Final Answer:
The final displacement of the bird so that it reaches the origin is:
\[ S_3 = -46\hat{i} - 65\hat{j} + 65\hat{k} \]