What displacement must be added to the displacement `25hat(i)-6hat(j)`m to give a displacement of 7.0 m pointing in the x-direction ?

Options

A. `18hat(i)-6hat(j)`
B. `32hat(i)-13hat(j)`
C. `-18hat(i)+6hat(j)`
D. `-25hat(i)+13hat(j)`

Explanation

To solve the problem, we need to find the displacement vector \( \mathbf{x} \) that must be added to the given displacement vector \( \mathbf{d_1} = 25\hat{i} - 6\hat{j} \) m to result in a new displacement vector \( \mathbf{d_2} = 7.0\hat{i} \) m, which points in the x-direction.

### Step-by-Step Solution:

1. **Identify the Given Vectors**:
- The initial displacement vector is:
\[ \mathbf{d_1} = 25\hat{i} - 6\hat{j} \]
- The desired displacement vector is:
\[ \mathbf{d_2} = 7.0\hat{i} \]

2. **Set Up the Equation**:
- We need to find the displacement vector \( \mathbf{x} \) such that:
\[ \mathbf{d_1} + \mathbf{x} = \mathbf{d_2} \]
- Rearranging this gives:
\[ \mathbf{x} = \mathbf{d_2} - \mathbf{d_1} \]

3. **Substitute the Vectors**:
- Substitute \( \mathbf{d_1} \) and \( \mathbf{d_2} \) into the equation:
\[ \mathbf{x} = (7.0\hat{i}) - (25\hat{i} - 6\hat{j}) \]

4. **Simplify the Expression**:
- Distributing the negative sign:
\[ \mathbf{x} = 7.0\hat{i} - 25\hat{i} + 6\hat{j} \]
- Combine like terms:
\[ \mathbf{x} = (7.0 - 25)\hat{i} + 6\hat{j} = -18\hat{i} + 6\hat{j} \]

5. **Final Result**:
- The displacement that must be added is:
\[ \mathbf{x} = -18\hat{i} + 6\hat{j} \text{ m} \]

### Summary:
To achieve a displacement of \( 7.0 \hat{i} \) m from the initial displacement \( 25\hat{i} - 6\hat{j} \) m, we need to add the displacement vector \( -18\hat{i} + 6\hat{j} \) m.