What should be added in vector `a=3hati+4hatj-2hatk` to get its resultant a unit vector `hati`?

Options

A. `-2hati-4hatj+2hatk`
B. `-2hati+4hatj-2hatk`
C. `2hati+4hatj-2hatk`
D. none of these

Explanation

To solve the problem, we need to find the vector \( \mathbf{x} \) that should be added to the vector \( \mathbf{a} = 3\hat{i} + 4\hat{j} - 2\hat{k} \) in order to obtain the resultant vector \( \hat{i} \).

### Step-by-Step Solution:

1. **Set up the equation**: We want to find \( \mathbf{x} \) such that:
\[ \mathbf{a} + \mathbf{x} = \hat{i} \]

2. **Substitute the value of \( \mathbf{a} \)**: We know that \( \mathbf{a} = 3\hat{i} + 4\hat{j} - 2\hat{k} \). Substituting this into the equation gives:
\[ (3\hat{i} + 4\hat{j} - 2\hat{k}) + \mathbf{x} = \hat{i} \]

3. **Rearrange the equation**: To isolate \( \mathbf{x} \), we rearrange the equation:
\[ \mathbf{x} = \hat{i} - (3\hat{i} + 4\hat{j} - 2\hat{k}) \]

4. **Distribute the negative sign**: This simplifies to:
\[ \mathbf{x} = \hat{i} - 3\hat{i} - 4\hat{j} + 2\hat{k} \]

5. **Combine like terms**: Now, we combine the \( \hat{i} \) components:
\[ \mathbf{x} = (1 - 3)\hat{i} - 4\hat{j} + 2\hat{k} \]
\[ \mathbf{x} = -2\hat{i} - 4\hat{j} + 2\hat{k} \]

6. **Final result**: Therefore, the vector that should be added to \( \mathbf{a} \) to obtain the resultant \( \hat{i} \) is:
\[ \mathbf{x} = -2\hat{i} - 4\hat{j} + 2\hat{k} \]

### Summary:
The vector \( \mathbf{x} \) that needs to be added to \( \mathbf{a} \) is:
\[ \mathbf{x} = -2\hat{i} - 4\hat{j} + 2\hat{k} \]