Following forces start acting on a particle at rest at the origin of the co-ordiante system simultaneously
`vecF_1=-4hati-4hatj+5hatk`,`vacF_2=5hati+8hatj+6hatk`,`vecF_3=-3hati+4hatj-7hatk` and `vecF_4=2hati-3hatj-2hatk` then the particle will move

Options

A. in x-y plane
B. in y-z plane
C. in x-z plane
D. along x-axis

Explanation

To solve the problem of determining the direction in which the particle will move when subjected to multiple forces, we need to find the resultant force acting on the particle. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Forces
The forces acting on the particle are given as:
- \(\vec{F_1} = -4\hat{i} - 4\hat{j} + 5\hat{k}\)
- \(\vec{F_2} = 5\hat{i} + 8\hat{j} + 6\hat{k}\)
- \(\vec{F_3} = -3\hat{i} + 4\hat{j} - 7\hat{k}\)
- \(\vec{F_4} = 2\hat{i} - 3\hat{j} - 2\hat{k}\)

### Step 2: Write the Resultant Force Equation
The resultant force \(\vec{F_r}\) is the vector sum of all the individual forces:
\[ \vec{F_r} = \vec{F_1} + \vec{F_2} + \vec{F_3} + \vec{F_4} \]

### Step 3: Substitute the Values
Substituting the values of the forces into the equation:
\[ \vec{F_r} = (-4\hat{i} - 4\hat{j} + 5\hat{k}) + (5\hat{i} + 8\hat{j} + 6\hat{k}) + (-3\hat{i} + 4\hat{j} - 7\hat{k}) + (2\hat{i} - 3\hat{j} - 2\hat{k}) \]

### Step 4: Combine Like Terms
Now, we will combine the components along the \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) directions separately:

- **For the \(\hat{i}\) component:**
\[ -4 + 5 - 3 + 2 = 0 \]

- **For the \(\hat{j}\) component:**
\[ -4 + 8 + 4 - 3 = 5 \]

- **For the \(\hat{k}\) component:**
\[ 5 + 6 - 7 - 2 = 2 \]

### Step 5: Write the Resultant Force
Now we can write the resultant force:
\[ \vec{F_r} = 0\hat{i} + 5\hat{j} + 2\hat{k} \]

### Step 6: Determine the Direction of Motion
The resultant force \(\vec{F_r} = 0\hat{i} + 5\hat{j} + 2\hat{k}\) indicates that there is no net force in the \(\hat{i}\) direction (x-axis), but there are positive components in the \(\hat{j}\) (y-axis) and \(\hat{k}\) (z-axis) directions. Therefore, the particle will move in the YZ plane.

### Final Answer
The particle will move in the YZ plane.

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