A body is moving under the action of two force `vec(F_(1)=2hati-5hatj` , `vec(F_(2)=3hati-4hatj` . Its velocity will become uniform under a third force `vec(F_(3)` given by.

Options

A. `5hati-9hatj`
B. `-5hati-9hatj`
C. `5hati+9hatj`
D. `-5hati+9hatj`

Answer

To find the third force \( \vec{F_3} \) that will make the velocity of the body uniform, we need to consider the forces acting on the body. The body is currently under the influence of two forces \( \vec{F_1} \) and \( \vec{F_2} \).

### Step 1: Identify the given forces
The forces are given as:
\[ \vec{F_1} = 2\hat{i} - 5\hat{j} \]
\[ \vec{F_2} = 3\hat{i} - 4\hat{j} \]

### Step 2: Calculate the resultant force
To find the resultant of the two forces, we add them together:
\[ \vec{F_{total}} = \vec{F_1} + \vec{F_2} \]
Substituting the values:
\[ \vec{F_{total}} = (2\hat{i} - 5\hat{j}) + (3\hat{i} - 4\hat{j}) \]
Combine the components:
\[ \vec{F_{total}} = (2 + 3)\hat{i} + (-5 - 4)\hat{j} = 5\hat{i} - 9\hat{j} \]

### Step 3: Determine the third force
For the velocity to become uniform, the third force \( \vec{F_3} \) must be equal and opposite to the resultant force \( \vec{F_{total}} \):
\[ \vec{F_3} = -\vec{F_{total}} = - (5\hat{i} - 9\hat{j}) \]
Thus:
\[ \vec{F_3} = -5\hat{i} + 9\hat{j} \]

### Final Answer
The third force \( \vec{F_3} \) that will make the velocity of the body uniform is:
\[ \vec{F_3} = -5\hat{i} + 9\hat{j} \]