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 determine the vector \( \vec{M} \) that, when added to the resultant vector \( \vec{R} \) of \( \vec{P} \) and \( \vec{Q} \), gives a unit vector along the X-axis.

### 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 \( \vec{R} \)**:
\[ \vec{R} = \vec{P} + \vec{Q} \]
\[ \vec{R} = (2\hat{i} + 7\hat{j} - 10\hat{k}) + (\hat{i} + 2\hat{j} + 3\hat{k}) \]
\[ \vec{R} = (2 + 1)\hat{i} + (7 + 2)\hat{j} + (-10 + 3)\hat{k} \]
\[ \vec{R} = 3\hat{i} + 9\hat{j} - 7\hat{k} \]

3. **Determine the unit vector along the X-axis**:
The unit vector along the X-axis is:
\[ \hat{u} = \hat{i} \]

4. **Set up the equation**:
We need to find \( \vec{M} \) such that:
\[ \vec{R} + \vec{M} = \hat{u} \]
Substituting \( \vec{R} \):
\[ 3\hat{i} + 9\hat{j} - 7\hat{k} + \vec{M} = \hat{i} \]

5. **Rearranging the equation to find \( \vec{M} \)**:
\[ \vec{M} = \hat{i} - (3\hat{i} + 9\hat{j} - 7\hat{k}) \]
\[ \vec{M} = \hat{i} - 3\hat{i} - 9\hat{j} + 7\hat{k} \]
\[ \vec{M} = (1 - 3)\hat{i} - 9\hat{j} + 7\hat{k} \]
\[ \vec{M} = -2\hat{i} - 9\hat{j} + 7\hat{k} \]

### Final Answer:
The vector \( \vec{M} \) is:
\[ \vec{M} = -2\hat{i} - 9\hat{j} + 7\hat{k} \]