Given:
Force, F = 3^i c^j 2^k
Displacement, S = -4^i 2^j 3^k
Work done, W = 6 J

To Find:
The value of c.

Solution:
1. Calculation of Work Done:
The work done is given by the dot product of force and displacement:
W = F · S

The dot product of two vectors is given by:
A · B = |A| |B| cosθ

where |A| and |B| are the magnitudes of vectors A and B, and θ is the angle between them.

In this case, the dot product can be written as:
W = |F| |S| cosθ

Since both force and displacement vectors only have positive components, the angle between them is 0 degrees, and cos0° = 1. Therefore, the equation simplifies to:
W = |F| |S|

The magnitudes of the vectors can be calculated as follows:
|F| = sqrt(3^2 + c^2 + 2^2) = sqrt(9 + c^2 + 4) = sqrt(c^2 + 13)
|S| = sqrt((-4)^2 + 2^2 + 3^2) = sqrt(16 + 4 + 9) = sqrt(29)

Substituting these values into the equation for work done:
6 = sqrt(c^2 + 13) * sqrt(29)

Squaring both sides of the equation:
36 = (c^2 + 13) * 29

Expanding the equation:
36 = 29c^2 + 377

Rearranging the equation:
29c^2 = 36 - 377
29c^2 = -341
c^2 = -341/29

Since the square of a real number cannot be negative, it is not possible to find a real value for c that satisfies this equation. Therefore, there is no real value of c that would make the work done equal to 6 J.

Conclusion:
There is no real value of c that satisfies the given conditions.