A body is at rest under the action of three forces, two of which are `vec(F)_(1)= 4hat(i), vec(F)_(2)= 6hat(j)`, the third force is

Options

A. `4hati+6hatj`
B. `4hati-6hatj`
C. `-4hati+6hatj`
D. `-4hati-6hatj`

Explanation

To find the third force acting on a body at rest under the action of three forces, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
A body is at rest, which means the net force acting on it must be zero. This is a consequence of Newton's First Law of Motion.

2. **Identify the Given Forces**:
We have two forces given:
- \( \vec{F_1} = 4 \hat{i} \)
- \( \vec{F_2} = 6 \hat{j} \)

3. **Set Up the Equation for Equilibrium**:
Since the body is at rest, the vector sum of all forces acting on it must equal zero:
\[ \vec{F_1} + \vec{F_2} + \vec{F_3} = \vec{0} \]

4. **Substituting the Known Forces**:
Substitute the values of \( \vec{F_1} \) and \( \vec{F_2} \) into the equation:
\[ 4 \hat{i} + 6 \hat{j} + \vec{F_3} = \vec{0} \]

5. **Rearranging the Equation**:
To find \( \vec{F_3} \), we rearrange the equation:
\[ \vec{F_3} = - (4 \hat{i} + 6 \hat{j}) \]

6. **Calculating \( \vec{F_3} \)**:
Simplifying gives us:
\[ \vec{F_3} = -4 \hat{i} - 6 \hat{j} \]

7. **Conclusion**:
The third force \( \vec{F_3} \) that keeps the body at rest is:
\[ \vec{F_3} = -4 \hat{i} - 6 \hat{j} \]

### Final Answer:
The third force is \( \vec{F_3} = -4 \hat{i} - 6 \hat{j} \).

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