A particle starts moving from the point (1, 2, 3) under velocity ` 1 hat I +2hat j + 3hatk ` for two seconds. After that it moves under velocity ` 1 hat I -2hat j + 3hatk ` for the next two seconds. What is the coordinate of the final points?

Options

A. ` (5,2,5) `
B. ` ( 5,2,15) `
C. `( 1,2,15)`
D. none of these

Explanation

To solve the problem step by step, we will calculate the displacement of the particle during each time interval and then find the final coordinates.

### Step 1: Identify the initial position and velocity
The particle starts at the point \( (1, 2, 3) \) and has an initial velocity given by:
\[ \mathbf{v_1} = \hat{i} + 2\hat{j} + 3\hat{k} \]

### Step 2: Calculate the displacement during the first 2 seconds
Using the formula for displacement:
\[ \text{Displacement} = \text{Velocity} \times \text{Time} \]
For the first 2 seconds:
\[ \mathbf{s_1} = \mathbf{v_1} \times t = (\hat{i} + 2\hat{j} + 3\hat{k}) \times 2 \]
Calculating this gives:
\[ \mathbf{s_1} = 2\hat{i} + 4\hat{j} + 6\hat{k} \]

### Step 3: Update the position after the first 2 seconds
The new position after the first 2 seconds can be calculated as:
\[ \mathbf{r_1} = \text{Initial Position} + \mathbf{s_1} \]
Substituting the values:
\[ \mathbf{r_1} = (1\hat{i} + 2\hat{j} + 3\hat{k}) + (2\hat{i} + 4\hat{j} + 6\hat{k}) = (1 + 2)\hat{i} + (2 + 4)\hat{j} + (3 + 6)\hat{k} \]
This simplifies to:
\[ \mathbf{r_1} = 3\hat{i} + 6\hat{j} + 9\hat{k} \]

### Step 4: Identify the second velocity
After the first 2 seconds, the particle moves under a new velocity:
\[ \mathbf{v_2} = \hat{i} - 2\hat{j} + 3\hat{k} \]

### Step 5: Calculate the displacement during the next 2 seconds
Using the same displacement formula:
\[ \mathbf{s_2} = \mathbf{v_2} \times t = (\hat{i} - 2\hat{j} + 3\hat{k}) \times 2 \]
Calculating this gives:
\[ \mathbf{s_2} = 2\hat{i} - 4\hat{j} + 6\hat{k} \]

### Step 6: Update the position after the next 2 seconds
The final position after the next 2 seconds can be calculated as:
\[ \mathbf{r_2} = \mathbf{r_1} + \mathbf{s_2} \]
Substituting the values:
\[ \mathbf{r_2} = (3\hat{i} + 6\hat{j} + 9\hat{k}) + (2\hat{i} - 4\hat{j} + 6\hat{k}) = (3 + 2)\hat{i} + (6 - 4)\hat{j} + (9 + 6)\hat{k} \]
This simplifies to:
\[ \mathbf{r_2} = 5\hat{i} + 2\hat{j} + 15\hat{k} \]

### Final Coordinates
Thus, the final coordinates of the particle are:
\[ (5, 2, 15) \]