# Find the work done by the force $f=\overrightarrow{2i} + \overrightarrow{j}+\overrightarrow{k}$ acting on particle, if the particle is displaced from the point with position vector $\overrightarrow{2i}+ \overrightarrow{2j}+ \overrightarrow{2k}$ to the point with the position vector $\overrightarrow{3i}+\overrightarrow{4j}+\overrightarrow{5k}$ .

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• The work done by a force $\overrightarrow F$ in displacing a particle through $\overrightarrow d$ is $\overrightarrow F.\overrightarrow d = |\overrightarrow F|$ ( displacement in the direction of $\overrightarrow F$)
The particle is displaced from $A ( O\overrightarrow A=2\overrightarrow i+2\overrightarrow j+2\overrightarrow k) \: to \: B(O\overrightarrow B = 3\overrightarrow i+4\overrightarrow j+5\overrightarrow k)$ by the force $\overrightarrow F = 2\overrightarrow i+\overrightarrow j+\overrightarrow k.$ The displacement is $\overrightarrow {AB} = \overrightarrow d=\overrightarrow {OB}-\overrightarrow {OA}=\overrightarrow i+2\overrightarrow j+3\overrightarrow k$
The work done by $\overrightarrow F, \: w = \overrightarrow F.\overrightarrow d = (2\overrightarrow i+\overrightarrow j+\overrightarrow k).(\overrightarrow i+2\overrightarrow j+3\overrightarrow k)$
$= (2)(1)+(1)(2)+(1)(3)$
$= 2+2+3 = 7$ units.