The vector that must be added to the vector `hati-4hatj+2hatk` and `3hati+3hatj-7hatk` so that the resultant vector is a unit vector along the y-axis is

Options

A. `4hati +2hatj +5hatk `
B. `-4hati +2hatj +5hatk `
C. `3hati +4hatj +5hatk `
D. Null vector

Explanation

To solve the problem, we need to find a vector \( \vec{C} \) that, when added to the vectors \( \vec{A} = \hat{i} - 4\hat{j} + 2\hat{k} \) and \( \vec{B} = 3\hat{i} + 3\hat{j} - 7\hat{k} \), results in a unit vector along the y-axis, which is represented as \( \hat{j} \).

### Step-by-Step Solution:

1. **Identify the vectors**:
- Let \( \vec{A} = \hat{i} - 4\hat{j} + 2\hat{k} \)
- Let \( \vec{B} = 3\hat{i} + 3\hat{j} - 7\hat{k} \)

2. **Add the vectors \( \vec{A} \) and \( \vec{B} \)**:
\[ \vec{A} + \vec{B} = (\hat{i} + 3\hat{i}) + (-4\hat{j} + 3\hat{j}) + (2\hat{k} - 7\hat{k}) \]
\[ = 4\hat{i} - \hat{j} - 5\hat{k} \]

3. **Set up the equation for the resultant vector**:
We want:
\[ \vec{A} + \vec{B} + \vec{C} = \hat{j} \]
Therefore, we can substitute:
\[ 4\hat{i} - \hat{j} - 5\hat{k} + \vec{C} = \hat{j} \]

4. **Rearranging the equation to find \( \vec{C} \)**:
\[ \vec{C} = \hat{j} - (4\hat{i} - \hat{j} - 5\hat{k}) \]
\[ = \hat{j} - 4\hat{i} + \hat{j} + 5\hat{k} \]
\[ = -4\hat{i} + 2\hat{j} + 5\hat{k} \]

5. **Final result**:
Therefore, the vector \( \vec{C} \) that must be added is:
\[ \vec{C} = -4\hat{i} + 2\hat{j} + 5\hat{k} \]

### Conclusion:
The vector that must be added is \( -4\hat{i} + 2\hat{j} + 5\hat{k} \).