Final answer:
The conditions for the line integral to be negative are: a < b, c > d, and e < f.
Step-by-step explanation:
To determine the conditions on a, b, c, d, e, and f so that the line integral of the vector field on the curve is negative, we need to consider the properties of the vector field and the direction of the curve.
One condition for the line integral to be negative is that the vector field points in the opposite direction of the curve. This means that the dot product of the vector field and the tangent vector to the curve should be negative.
Another condition is that the magnitudes of the vector field and the tangent vector should be different. If the magnitudes are the same, the dot product will always be positive or zero.
Therefore, the conditions for the line integral to be negative are: a < b, c > d, and e < f.