Assume that a particle is displaced from point A = (2, - 6, 1) to the point B = (5, 9, 7) under the influence of force \(\vec F = \;2\hat i + \hat j + \hat k\). Find the work done in displacing the particle from point A to B.

Concept:

I. If a particle is displace from point A to B under the influence of force \(\vec F\). Then the work done in displacing the particle from point A to B is given by: \(W = \;\vec F \cdot \vec d\)

II. If \(\vec a = {a_1}\hat i + {a_2}\hat j + {a_3}\hat k\;and\;\vec b = {b_1}\hat i + {b_2}\hat j + {b_3}\hat k\) then \(\vec a \cdot \;\vec b = {a_1}{b_1} + {a_2}{b_2} + {a_3}{b_3}\)

Calculation:

Given: Particle is displace from point A = (2, - 6, 1) to the point B = (5, 9, 7) under the influence of force \(\vec F = \;2\hat i + \hat j + \hat k\)

So, the displacement of particle is given by:

\(\vec d = \;\overrightarrow {AB} = \left( {5\hat i + 9\hat j + 7\hat k} \right) - \left( {2\hat i - 6\hat j + \hat k} \right) = \;3\hat i + 15\hat j + 6\hat k\)

As we know that, If a particle is displace from point A to B under the influence of force \(\vec F\). Then the work done in displacing the particle from point A to B is given by: \(W = \;\vec F \cdot \vec d\)

\(W = \;\vec F \cdot \vec d = \left( {2\hat i + \hat j + \hat k} \right) \cdot \left( {3\hat i + 15\hat j + 6\hat k} \right) = 6 + 15 + 6 = 27\;units\)