Complimentary shear stress in an element:
In mechanics, shear stress is defined as the force per unit area acting parallel to a given plane within a material. When a material is subjected to shear stress, it deforms by shearing or sliding along the plane.

In a solid element, such as a beam or a plate, shear stress can be resolved into two components - the shear stress acting in one direction and the complimentary shear stress acting in the perpendicular direction. The complimentary shear stress is the shear stress acting in the opposite direction to the original shear stress.

Options:
a) Equal in sign: This option suggests that the complimentary shear stress has the same sign as the original shear stress. However, this is not always the case.

b) Equal in both magnitude and sign: This option suggests that the complimentary shear stress has the same magnitude and sign as the original shear stress. Again, this is not always true.

c) Equal in magnitude but opposite in sign: This option suggests that the complimentary shear stress has the same magnitude as the original shear stress but acts in the opposite direction. This is the correct answer.

Explanation:
When a solid element is subjected to shear stress, the material deforms by shearing along a plane. This shearing causes the material to experience internal forces that resist the deformation. These internal forces can be resolved into two components - one acting in the direction of the original shear stress and the other acting in the perpendicular direction.

The complimentary shear stress is the shear stress acting in the opposite direction to the original shear stress. It has the same magnitude as the original shear stress but acts in the perpendicular direction. This is due to the conservation of momentum and Newton's third law of motion.

For example, if a beam is subjected to shear stress in the horizontal direction, the complimentary shear stress will act in the vertical direction. If the original shear stress is positive, indicating a shearing force in the positive x-direction, the complimentary shear stress will be negative, indicating a shearing force in the negative y-direction.

In conclusion, the complimentary shear stress in an element is always equal in magnitude but opposite in sign to the original shear stress.