Why is the gradient of F a constant multiple of a parallel vector?

Question

Problem 14:

Solution:

I don't understand why the gradient of F is a constant multiple of a parallel vector. Why is the equation in the black box true? I understand why it would be true if the gradient of F is a stretched version of (27, 8, 1) but it is only parallel to (27, 8, 1). Doesn't that mean that there is a possibility that there is some sort of translation?

Answer 1

The question is explicitly asking you to find where the gradient is parallel to the given vector. Two vectors are parallel if and only if they are multiples of each other. These are not "free vectors" but rather vectors considered in $\mathbb{R^3}$, so they all begin at the origin.