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

Answer

To solve the problem, we need to find the displacement vector that must be added to the given displacement vector \( \mathbf{s} = 25\hat{i} - 6\hat{j} \) m to achieve a resultant displacement of \( 7\hat{i} \) m, which points in the x-direction.

### Step-by-step Solution:

1. **Identify the Given Displacement Vector**:
The given displacement vector is:
\[ \mathbf{s} = 25\hat{i} - 6\hat{j} \]

2. **Identify the Required Resultant Displacement Vector**:
The required resultant displacement vector is:
\[ \mathbf{r} = 7\hat{i} \]

3. **Set Up the Equation**:
Let the displacement vector that needs to be added be \( \mathbf{p} \). According to vector addition, we have:
\[ \mathbf{s} + \mathbf{p} = \mathbf{r} \]
Substituting the known vectors:
\[ (25\hat{i} - 6\hat{j}) + \mathbf{p} = 7\hat{i} \]

4. **Rearranging the Equation**:
To find \( \mathbf{p} \), we can rearrange the equation:
\[ \mathbf{p} = \mathbf{r} - \mathbf{s} \]
Substituting the values:
\[ \mathbf{p} = 7\hat{i} - (25\hat{i} - 6\hat{j}) \]

5. **Simplifying the Expression**:
Distributing the negative sign:
\[ \mathbf{p} = 7\hat{i} - 25\hat{i} + 6\hat{j} \]
This simplifies to:
\[ \mathbf{p} = (7 - 25)\hat{i} + 6\hat{j} \]
\[ \mathbf{p} = -18\hat{i} + 6\hat{j} \]

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