Following forces start acting on a particle at rest at the origin of the co-ordinate system simultaneously `vec(F)_(1)= -4hat(i)-5hat(j)+5hat(k), vec(F)_(2)= 5hat(i)+8hat(j)+6hat(k), vec(F)_(3)= -3hat(i)+4hat(j)-7hat(k)` and `vec(F)_(4)= 2hat(i)-3hat(k)` 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, we need to find the net force acting on the particle by summing the individual forces. The forces are given as vectors, and we will add them component-wise.

### Step-by-Step Solution:

1. **Identify the Forces**:
- \( \vec{F}_1 = -4\hat{i} - 5\hat{j} + 5\hat{k} \)
- \( \vec{F}_2 = 5\hat{i} + 8\hat{j} + 6\hat{k} \)
- \( \vec{F}_3 = -3\hat{i} + 4\hat{j} - 7\hat{k} \)
- \( \vec{F}_4 = 2\hat{i} - 3\hat{k} \)

2. **Sum the Forces**:
We will add the forces component-wise (i.e., add all the \( \hat{i} \) components together, all the \( \hat{j} \) components together, and all the \( \hat{k} \) components together).

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

- **For the \( \hat{j} \) component**:
\[ F_{net, j} = -5 + 8 + 4 = 7 \]

- **For the \( \hat{k} \) component**:
\[ F_{net, k} = 5 + 6 - 7 - 3 = 1 \]

3. **Write the Net Force**:
Now we can write the net force vector:
\[ \vec{F}_{net} = 0\hat{i} + 7\hat{j} + 1\hat{k} \]

4. **Determine the Direction of Motion**:
Since the net force is not zero, the particle will move in the direction of the net force. The net force vector indicates that the particle will move in the positive \( \hat{j} \) direction (upward) and slightly in the positive \( \hat{k} \) direction (forward).

5. **Conclusion**:
The particle will move in the direction of the vector \( 7\hat{j} + 1\hat{k} \).