Final answer:
To determine if the work done by a vector field is positive, negative, or zero, one must consider the angle between the force vector and the displacement vector. Positive work is achieved when they are in the same direction, negative when in opposite directions, and zero when they are perpendicular or there is no displacement.
Step-by-step explanation:
Understanding Work Done by a Force in a Vector Field
To determine if the work done by a vector field is positive, negative, or zero, we need to consider the relationship between the force vector and the displacement of an object. The work done is calculated as the dot product between the force vector and the displacement vector, which is equivalent to the product of the magnitude of the force, the magnitude of the displacement, and the cosine of the angle between them. If the force is in the same direction as the displacement, the angle is between 0° and 90°, the cosine of the angle is positive, and thus the work done is positive. Conversely, if the force is in the opposite direction to the displacement, the angle is between 90° and 180°, the cosine is negative, and the work done is negative. If the force is perpendicular to the displacement, or if there is no displacement, then no work is done, as the cosine of 90° is zero.
In summary, positive work is done when the force and displacement are in the same direction, negative work is done when they are in opposite directions, and zero work is done when the force and displacement are perpendicular to each other or when displacement is zero.