Determine that vector which when added to the resultant of `vecP=2hati+7hatj-10hatkandvecQ=hati+2hatj+3hatk` gives a unit vector along X-axis.

Explanation

To solve the problem, we need to find a vector \( \vec{R} \) such that when it is added to the resultant of vectors \( \vec{P} \) and \( \vec{Q} \), it results in a unit vector along the X-axis, which is represented by \( \hat{i} \).

### Step-by-Step Solution:

1. **Identify the vectors:**
- Given \( \vec{P} = 2\hat{i} + 7\hat{j} - 10\hat{k} \)
- Given \( \vec{Q} = \hat{i} + 2\hat{j} + 3\hat{k} \)

2. **Calculate the resultant vector \( \vec{R_{PQ}} \):**
- \( \vec{R_{PQ}} = \vec{P} + \vec{Q} \)
- \( \vec{R_{PQ}} = (2\hat{i} + 7\hat{j} - 10\hat{k}) + (\hat{i} + 2\hat{j} + 3\hat{k}) \)
- Combine like terms:
\[ \vec{R_{PQ}} = (2 + 1)\hat{i} + (7 + 2)\hat{j} + (-10 + 3)\hat{k} \]
\[ \vec{R_{PQ}} = 3\hat{i} + 9\hat{j} - 7\hat{k} \]

3. **Set up the equation:**
- We want \( \vec{R} + \vec{R_{PQ}} = \hat{i} \)
- Substitute \( \vec{R_{PQ}} \):
\[ \vec{R} + (3\hat{i} + 9\hat{j} - 7\hat{k}) = \hat{i} \]

4. **Isolate \( \vec{R} \):**
- Rearranging gives:
\[ \vec{R} = \hat{i} - (3\hat{i} + 9\hat{j} - 7\hat{k}) \]
- Distributing the negative sign:
\[ \vec{R} = \hat{i} - 3\hat{i} - 9\hat{j} + 7\hat{k} \]
- Combine like terms:
\[ \vec{R} = (1 - 3)\hat{i} - 9\hat{j} + 7\hat{k} \]
\[ \vec{R} = -2\hat{i} - 9\hat{j} + 7\hat{k} \]

5. **Final Result:**
- The required vector \( \vec{R} \) is:
\[ \vec{R} = -2\hat{i} - 9\hat{j} + 7\hat{k} \]