Mathematics

When you divide a vector by a scalar, you scale all its components by that same factor. The magnitude of a vector is just a number—the "length" you're dividing by. So dividing $\vec{v}$ by $|\vec{v}|$ shrinks every component by exactly the factor needed to make the total length equal to 1. The direction stays unchanged because you're scaling uniformly, but the new magnitude becomes $\frac{|\vec{v}|}{|\vec{v}|} = 1$.

When you divide a vector by a scalar, you scale all its components by that same factor. The magnitude of a vector is just a number—the "length" you're dividing by. So dividing $\vec{v}$ by $|\vec{v}|$ shrinks every component by exactly the factor needed to make the total length equal to 1. The direction stays unchanged because you're scaling uniformly, but the new magnitude becomes $\frac{|\vec{v}|}{|\vec{v}|} = 1$.