A vector parallel to the vector `(hati+2hatj)` and having magnitude `3sqrt(5)` units is

Options

A. `3hati+6hatj`
B. `6hati-3hatj`
C. `4hati-2hatj`
D. `hati-2hatj`

Explanation

To find a vector parallel to the vector \( \hat{i} + 2\hat{j} \) with a magnitude of \( 3\sqrt{5} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Vector**:
The given vector is \( \mathbf{v} = \hat{i} + 2\hat{j} \).

2. **Find the Magnitude of the Given Vector**:
The magnitude of the vector \( \mathbf{v} \) is calculated using the formula:
\[ |\mathbf{v}| = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \]

3. **Determine the Unit Vector**:
To find the unit vector in the direction of \( \mathbf{v} \), we divide the vector by its magnitude:
\[ \hat{u} = \frac{\mathbf{v}}{|\mathbf{v}|} = \frac{\hat{i} + 2\hat{j}}{\sqrt{5}} = \frac{1}{\sqrt{5}}\hat{i} + \frac{2}{\sqrt{5}}\hat{j} \]

4. **Calculate the Required Vector**:
The required vector \( \mathbf{A} \) with a magnitude of \( 3\sqrt{5} \) in the direction of \( \hat{u} \) is given by:
\[ \mathbf{A} = |\mathbf{A}| \cdot \hat{u} = 3\sqrt{5} \left( \frac{1}{\sqrt{5}}\hat{i} + \frac{2}{\sqrt{5}}\hat{j} \right) \]

5. **Simplify the Expression**:
Simplifying the above expression:
\[ \mathbf{A} = 3\sqrt{5} \cdot \frac{1}{\sqrt{5}}\hat{i} + 3\sqrt{5} \cdot \frac{2}{\sqrt{5}}\hat{j} = 3\hat{i} + 6\hat{j} \]

6. **Final Result**:
Therefore, the vector parallel to \( \hat{i} + 2\hat{j} \) with a magnitude of \( 3\sqrt{5} \) is:
\[ \mathbf{A} = 3\hat{i} + 6\hat{j} \]