The displacement of a particle from a point having position vector ` 2hati + 4hatj` to another point having position vector ` 5hatj + 1hatj` is

Options

A. 3 units
B. `3sqrt2` units
C. 5 units
D. ` 5sqrt3`units

Answer

To find the displacement of a particle from one point to another in a plane, we can follow these steps:

### Step 1: Identify the position vectors
We have two points with their position vectors:
- Initial position vector \( \mathbf{r_1} = 2\hat{i} + 4\hat{j} \)
- Final position vector \( \mathbf{r_2} = 5\hat{i} + 1\hat{j} \)

### Step 2: Write the formula for displacement
The displacement vector \( \mathbf{s} \) is given by the difference between the final position vector and the initial position vector:
\[ \mathbf{s} = \mathbf{r_2} - \mathbf{r_1} \]

### Step 3: Substitute the position vectors
Substituting the values of \( \mathbf{r_1} \) and \( \mathbf{r_2} \):
\[ \mathbf{s} = (5\hat{i} + 1\hat{j}) - (2\hat{i} + 4\hat{j}) \]

### Step 4: Perform the subtraction
Now, we perform the subtraction component-wise:
\[ \mathbf{s} = (5\hat{i} - 2\hat{i}) + (1\hat{j} - 4\hat{j}) \]
\[ \mathbf{s} = 3\hat{i} - 3\hat{j} \]

### Step 5: Write the final displacement vector
Thus, the displacement vector is:
\[ \mathbf{s} = 3\hat{i} - 3\hat{j} \]

### Step 6: Calculate the magnitude of the displacement
The magnitude of the displacement vector can be calculated using the formula:
\[ |\mathbf{s}| = \sqrt{(3)^2 + (-3)^2} \]
\[ |\mathbf{s}| = \sqrt{9 + 9} = \sqrt{18} \]
\[ |\mathbf{s}| = 3\sqrt{2} \]

### Final Answer
The displacement of the particle is \( \mathbf{s} = 3\hat{i} - 3\hat{j} \) with a magnitude of \( 3\sqrt{2} \) units.

---