Question

Following forces start acting on a particle at rest at the origin of the coordinate system simultaneously $ {\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,{\text{and }}{\vec F_4} = 12\hat i - 3\hat j - 2\hat k $ then the particle will move
A) $ {\text{in x - y plane}} $
B) $ {\text{in y - z plane}} $
C) $ {\text{in x - z plane}} $
D) $ {\text{along x - axis}} $

Answer 1

Hint
In the question, the forces acting on a particle are given. By substituting those values in the force equation, we get the value of the total forces and the direction of the particle has been calculated.
The expression for find the forces is
 $\Rightarrow {F_{net}} = {\vec F_1} + {\vec F_2} + {\vec F_3} + {\vec F_4} $
Where,
 $\Rightarrow {\vec F_1} $ be the first force, $ {\vec F_2} $ be the second force, $ {\vec F_3} $ be the third force, $ {\vec F_4} $ be the fourth force and $ {F_{net}} $ be the total force acting on a particle.

Complete step by step answer
Given that
 $\Rightarrow {\vec F_1} = - 4\hat i - 5\hat j + 5\hat k $
 $\Rightarrow {\vec F_2} = - 5\hat i + 8\hat j + 6\hat k $
 $\Rightarrow {\vec F_3} = - 3\hat i + 4\hat j - 7\hat k $
 $\Rightarrow {\vec F_4} = 12\hat i - 3\hat j - 2\hat k $
Following forces acting on a particle at rest at the origin. So, the forces acting on the particle in the coordinate system simultaneously. Then we can find that the particle moves in the direction of plane we get,
Total force $ {F_{net}} = {\vec F_1} + {\vec F_2} + {\vec F_3} + {\vec F_4} $
Substitute the known values in the above equation of force, we get
 $\Rightarrow {F_{net}} = \left( { - 4\hat i - 5\hat j + 5\hat k} \right) + \left( { - 5\hat i + 8\hat j + 6\hat k} \right) + \left( { - 3\hat i + 4\hat j - 7\hat k} \right) + \left( {12\hat i - 3\hat j - 2\hat k} \right) $
Performing the arithmetic operation in the above equation, we get
 $\Rightarrow {F_{net}} = 0 + 4\hat j + 2\hat k $
Here the direction of the x axis will be zero. So, it moves in the y and z axes.
Therefore, the particle will move in the $ y - z\,{\text{plane}}{\text{.}} $
Hence, from the above options, option (B) is correct.

Note
In the question, all the following forces are acting on a particle. But here all the following forces acting on a particle, is at rest all the force acting on the particle simultaneously. By using the arithmetic operations, we get the direction of the forces.