Three forces given by vectors ` 2hati + 2hatj. 2hati -2hatj and - 4hatj` are acting together on a point object at rest. The object moves along the direction.

Options

A. x -axis
B. y-axis
C. z-axis
D. object does not move

Answer

To solve the problem step by step, we need to find the resultant force acting on the point object and determine the direction of its motion.

### Step 1: Identify the Forces
We have three forces acting on the object:
- \( \mathbf{F_1} = 2\hat{i} + 2\hat{j} \)
- \( \mathbf{F_2} = 2\hat{i} - 2\hat{j} \)
- \( \mathbf{F_3} = -4\hat{j} \)

### Step 2: Calculate the Net Force
The net force \( \mathbf{F_{net}} \) is the vector sum of all the forces acting on the object:
\[ \mathbf{F_{net}} = \mathbf{F_1} + \mathbf{F_2} + \mathbf{F_3} \]
Substituting the values of the forces:
\[ \mathbf{F_{net}} = (2\hat{i} + 2\hat{j}) + (2\hat{i} - 2\hat{j}) + (-4\hat{j}) \]

### Step 3: Combine Like Terms
Now, we combine the \( \hat{i} \) and \( \hat{j} \) components separately:
- For the \( \hat{i} \) components:
\[ 2\hat{i} + 2\hat{i} = 4\hat{i} \]

- For the \( \hat{j} \) components:
\[ 2\hat{j} - 2\hat{j} - 4\hat{j} = 0 - 4\hat{j} = -4\hat{j} \]

### Step 4: Write the Resultant Force
Now we can write the resultant force:
\[ \mathbf{F_{net}} = 4\hat{i} - 4\hat{j} \]

### Step 5: Determine the Direction of Motion
The object is at rest initially, and it will move in the direction of the resultant force. The resultant force has components in both the \( \hat{i} \) (x-axis) and \( \hat{j} \) (y-axis) directions.

To find the direction of motion, we can analyze the components:
- The positive \( \hat{i} \) component indicates motion in the positive x-direction.
- The negative \( \hat{j} \) component indicates motion in the negative y-direction.

### Step 6: Conclusion
The object will move in the direction of the vector \( 4\hat{i} - 4\hat{j} \). This means it will move diagonally in the plane, specifically in the direction that can be represented as:
\[ \text{Direction} = \tan^{-1}\left(\frac{-4}{4}\right) = \tan^{-1}(-1) \]
This corresponds to an angle of \( -45^\circ \) from the positive x-axis, indicating the object moves downwards to the right.

### Final Answer
The object will move in the direction of the vector \( 4\hat{i} - 4\hat{j} \).

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