Question

Which of the following quantities are vectors?

a) div(f(xyz))
b) grad(f(xyz))
c) u
d) All of the above

Answer 1

After evaluating the provided options, div(f(xyz)) is a scalar, grad(f(xyz)) is a vector, and 'u' is unspecified as a vector or scalar without further context. The only option containing two vectors and a scalar is (b) displacement, velocity, acceleration. Moreover, force is considered a vector quantity.

Step-by-step explanation:

To identify whether the quantities are vectors or scalars, we need to define what each of them represents. A vector is a quantity that has both magnitude and direction, whereas a scalar has only magnitude. From the provided options:

• div(f(xyz)) is a scalar; it represents the divergence of a vector field which is a scalar quantity.
• grad(f(xyz)) is a vector; it represents the gradient which gives the direction and rate of the fastest increase of a scalar field.
• u is unspecified here, but if 'u' represents velocity or displacement, it would be a vector; however, without context, we cannot confirm the nature of 'u'.

Now, considering the provided list and what defines a vector:

• distance is a scalar quantity as it has magnitude only.
• acceleration is a vector quantity because it has both magnitude and direction.
• speed is a scalar quantity as it represents the magnitude of velocity but without the direction.

From the choices given:

1. distance, acceleration, speed contains two scalars (distance, speed) and one vector (acceleration).
2. displacement, velocity, acceleration contains three vectors, as these all have magnitude and direction.
3. distance, mass, speed contains two scalars (distance, speed) and another scalar (mass).
4. displacement, speed, velocity contains two vectors (displacement, velocity) and one scalar (speed).

Therefore, the correct answer is:

1. (b) displacement, velocity, acceleration

Force, mentioned in option (3), is indeed a vector quantity because it has both magnitude and direction associated with it. Finally, the products of vectors, such as the scalar product (also called the dot product), is a scalar quantity because it results in a single number, thus not requiring direction.