unit vector conceptual

Question

For the following, is it always going to be equal to 1? $$\left|\frac{\vec{r}}{\vec{r}}\right|$$

Answer 1

You seem to have a few minus sign errors in your intermediate steps. But the final answer of $1$ is correct, and that is always going to be the case. i.e it is always true that for $\mathbf{r} \neq 0$, $\left\lVert \dfrac{\mathbf{r}}{\lVert \mathbf{r} \rVert} \right\rVert = \dfrac{\lVert \mathbf{r} \rVert}{ \lVert \mathbf{r} \rVert } = 1$.

Basically, it's saying that the length of the vector obtained by dividing the original vector by its length is $1$.