Theposition vector of a moving particle at time t is `r=3hati+4thatj-thatk` Its displacement during the time interval `t=1` s to t=3` s is

Options

A. `hatj-hatk`
B. `3hati+4hatj-hatk`
C. `9hati+36hatj-27hatk`
D. None of the above

Explanation

To find the displacement of the moving particle during the time interval from \( t = 1 \) s to \( t = 3 \) s, we will follow these steps:

### Step 1: Determine the position vector at \( t = 3 \) s
The position vector \( \mathbf{r} \) is given by:
\[ \mathbf{r} = 3\hat{i} + 4t\hat{j} - t\hat{k} \]
Substituting \( t = 3 \) s:
\[ \mathbf{r}_3 = 3\hat{i} + 4(3)\hat{j} - (3)\hat{k} \]
Calculating this gives:
\[ \mathbf{r}_3 = 3\hat{i} + 12\hat{j} - 3\hat{k} \]

### Step 2: Determine the position vector at \( t = 1 \) s
Now substituting \( t = 1 \) s into the position vector:
\[ \mathbf{r}_1 = 3\hat{i} + 4(1)\hat{j} - (1)\hat{k} \]
Calculating this gives:
\[ \mathbf{r}_1 = 3\hat{i} + 4\hat{j} - 1\hat{k} \]

### Step 3: Calculate the displacement
Displacement \( \mathbf{D} \) is given by the difference of the final and initial position vectors:
\[ \mathbf{D} = \mathbf{r}_3 - \mathbf{r}_1 \]
Substituting the values we calculated:
\[ \mathbf{D} = (3\hat{i} + 12\hat{j} - 3\hat{k}) - (3\hat{i} + 4\hat{j} - 1\hat{k}) \]
This simplifies to:
\[ \mathbf{D} = (3\hat{i} - 3\hat{i}) + (12\hat{j} - 4\hat{j}) + (-3\hat{k} + 1\hat{k}) \]
\[ \mathbf{D} = 0\hat{i} + 8\hat{j} - 2\hat{k} \]
Thus, the displacement vector is:
\[ \mathbf{D} = 8\hat{j} - 2\hat{k} \]

### Step 4: Check the options
Since the displacement \( 8\hat{j} - 2\hat{k} \) does not match any of the provided options, we conclude that the correct answer is option 4: none of these.

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