Forces 3î + 2ĵ + 5k̂ and 2î + ĵ - 3k̂ are acting on a particle and displace it from the point 2î - ĵ - 3k̂ to the point 4î - 3ĵ + 7k̂. The work done by the force is:

Concept:

• If two points A and B have position vectors  and  respectively, then the vector .

• For two vectors  and  at an angle θ to each other:
  • Dot Product is defined as: .
  • Resultant Vector is equal .

• Work: The work (W) done by a force () in moving (displacing) an object along a vector is given by: W = .

Calculation:

Let's say that the forces acting on the particle are  = 3î + 2ĵ + 5k̂ and  = 2î + ĵ - 3k̂.

∴ The resulting force acting on the particle will be .

⇒ = (3î + 2ĵ + 5k̂) + (2î + ĵ - 3k̂)

⇒  = 5î + 3ĵ + 2k̂.

Since the particle is moved from point 2î - ĵ - 3k̂ to the point 4î - 3ĵ + 7k̂, the displacement vector  will be:

 = (4î - 3ĵ + 7k̂) - (2î - ĵ - 3k̂)

⇒ ​ = 2î - 2ĵ + 10k̂.

And finally, the work done W will be:

W =  = (5î + 3ĵ + 2k̂).(2î - 2ĵ + 10k̂)

⇒ W = (5)(2) + (3)(-2) + (2)(10)

⇒ W = 10 - 6 + 20 = 24 units.