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

Options

A. `y=3-e^(-x)`
B. `y=1-4e^(-x)`
C. `y=1-3e^(-x)`
D. `y=3-4e^(-x)`

Explanation

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

### Step-by-step Solution:

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

2. **Calculate the Resultant Vector \( \vec{R}_{PQ} \):**
- The resultant vector \( \vec{R}_{PQ} \) is given by:
\[ \vec{R}_{PQ} = \vec{P} + \vec{Q} \]
- Substitute the values of \( \vec{P} \) and \( \vec{Q} \):
\[ \vec{R}_{PQ} = (2\hat{i} + 7\hat{j} - 10\hat{k}) + (\hat{i} + 2\hat{j} + 3\hat{k}) \]
- Combine like terms:
\[ \vec{R}_{PQ} = (2 + 1)\hat{i} + (7 + 2)\hat{j} + (-10 + 3)\hat{k} = 3\hat{i} + 9\hat{j} - 7\hat{k} \]

3. **Set Up the Equation for \( \vec{R} \):**
- We want \( \vec{R} + \vec{R}_{PQ} = \hat{i} \).
- Rearranging gives:
\[ \vec{R} = \hat{i} - \vec{R}_{PQ} \]

4. **Substitute \( \vec{R}_{PQ} \) into the Equation:**
- Substitute \( \vec{R}_{PQ} = 3\hat{i} + 9\hat{j} - 7\hat{k} \):
\[ \vec{R} = \hat{i} - (3\hat{i} + 9\hat{j} - 7\hat{k}) \]

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

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