A force \[\overrightarrow F = 3\widehat i + c\widehat j + 2\widehat k\] N acting on a particle causes a displacement \[\overrightarrow S = - 4\widehat i + 2\widehat j - 3\widehat k\] m. lf the work done is 6 Joule, then find the value of c.
A. 0
B. 1
C. 12
D. 6

Hint:We start proceeding with this problem by understanding the term displacement and work done. When a force is applied, the object changes its position which is known as displacement. Since it is a vector quantity it has both direction and magnitude. Work is nothing but a force needed to move an object from one place to another. The work-energy theorem states that work done on an object is equal to the change in the kinetic energy of an object.

Formula Used:
The formula to find the work done is given by,
\[W = \overrightarrow F \cdot \overrightarrow S \]
Where, \[\overrightarrow F \] is force applied on a particle and \[\overrightarrow S \] is displacement of a particle.

Complete step by step solution:
If a force \[\overrightarrow F = 6\widehat i + 2\widehat j - 3\widehat k\]acts on a particle by producing a displacement \[\overrightarrow S = 2\widehat i - 3\widehat j + x\widehat k\]. And the work done by this force is 6J, we need to find the value of c. Now by using the formula for work done we have,
\[W = \overrightarrow F \cdot \overrightarrow S \]
Here, \[\overrightarrow F \] is the force applied on a particle and \[\overrightarrow S \] is the displacement of a particle due to the force applied.

Now, substitute the values of force, displacement, and the work done which is provided in the data. Then we get,
\[\left( {3\widehat i + c\widehat j + 2\widehat k} \right) \cdot \left( { - 4\widehat i + 2\widehat j - 3\widehat k} \right) = 6\]
\[ \Rightarrow - 12 + 2c - 6 = 6\]
\[\Rightarrow 2c = 18 + 6\]
\[\Rightarrow 2c = 24\]
\[ \therefore c = 12\]
Therefore, the value of c is found to be 12.

Hence, option C is the correct answer.

Note:Don’t get confused with the notion of displacement, sometimes it is denoted by ‘S’ and sometimes by ‘d’. Work done can be positive, negative, or zero depending on the angle between the force applied and the displacement.