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

The assertion that the dot product guarantees an always nonnegative integrand is false, as it can be negative if vectors form an obtuse angle. It is false that only the perpendicular component of force contributes to work; in fact, it is the parallel component that does, making the last statement true.

Step-by-step explanation:

In physics, particularly when discussing work and forces, the dot product plays a critical role. The dot product is the result of the scalar multiplication of two vectors, leading to a scalar value, often used when calculating work done by a force over a certain displacement.

The first statement, which claims that the dot product ensures the integrand is always nonnegative, is false. The dot product can be negative if the angle between the vectors is greater than 90° but less than or equal to 180°.

The second statement, which suggests that only the component of the force perpendicular to the path contributes to the integral, is false. It's the component of the force that is parallel to the displacement that contributes to work.

The third statement, indicating that only the component of the force parallel to the path contributes to the integral, is true. When considering work done, it is indeed the parallel component of the force along the path of displacement that is pertinent.

Answer 2

The first two statements are false

The third statement is true

Step-by-step 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}| * |\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