Question

Now, consider whether the following statements are true or false: The dot product assures that the integrand is always nonnegative. The dot product indicates that only the component of the force perpendicular to the path contributes to the integral. The dot product indicates that only the component of the force parallel to the path contributes to the integral.

Answer 1

Answer:

The first two statements are false

The third statement is true

Explanation:

The dot product assures that the integrand is always nonnegative.

The dot product may be negative, this could ocurr when the vectors are directed oposite each other, for example take the unitary vector i and -i its doct product will give -1.

Another way to consider this is to take the definition of the dot product in terms of teh angle between the vetcors:

$\vec{A} \cdot \vec{B} = |\vec{A}| \times |\vec{B}| cos(\theta)$

When θ>π :

          cos(θ)<0

The dot product indicates that only the component of the force perpendicular to the path contributes to the integral

In fact the dot product is a projection of the vectors, the perpendicular component may be obtained using the cross product

The dot product indicates that only the component of the force parallel to the path contributes to the integral.

This one is true, since the dot product gives the projection of one vector to another, that is, the parallel component of the vector among the other one