If `vecA+vecB` is a unit vector along x-axis and `vecA=hati-hatj+hatk`, then what is `vecB`?

Explanation

To solve the problem, we need to find the vector \(\vec{B}\) given that \(\vec{A} + \vec{B}\) is a unit vector along the x-axis and \(\vec{A} = \hat{i} - \hat{j} + \hat{k}\).

### Step-by-Step Solution:

1. **Identify the Unit Vector Along the X-Axis**:
The unit vector along the x-axis is represented as:
\[ \hat{i} \]

2. **Set Up the Equation**:
According to the problem, we have:
\[ \vec{A} + \vec{B} = \hat{i} \]

3. **Rearrange to Find \(\vec{B}\)**:
We can rearrange the equation to express \(\vec{B}\) in terms of \(\vec{A}\):
\[ \vec{B} = \hat{i} - \vec{A} \]

4. **Substitute the Given Vector \(\vec{A}\)**:
We know that:
\[ \vec{A} = \hat{i} - \hat{j} + \hat{k} \]
Substituting this into the equation for \(\vec{B}\):
\[ \vec{B} = \hat{i} - (\hat{i} - \hat{j} + \hat{k}) \]

5. **Simplify the Expression**:
Distributing the negative sign:
\[ \vec{B} = \hat{i} - \hat{i} + \hat{j} - \hat{k} \]
The \(\hat{i}\) terms cancel out:
\[ \vec{B} = \hat{j} - \hat{k} \]

6. **Final Result**:
Thus, the vector \(\vec{B}\) is:
\[ \vec{B} = \hat{j} - \hat{k} \]