I've been trying to prove that $\left\Vert\dfrac{\vec{v}}{\Vert\vec{v}\Vert}\right\Vert=1, \quad \vec{v}\ne \vec{0}$. This is my attempt:  \begin{align} \vec{v}&amp;\in \mathbb{R}^n, \quad \vec{v}=\begin{pmatrix}v_1 \\ \vdots \\ v_n \end{pmatrix} \\ \Vert\vec{v}\Vert&amp;=\sqrt{\sum_{i=1}^nv_i^2} \\ \frac{\vec{v}}{\sqrt{\sum_{i=1}^nv_i^2}}&amp;=\begin{pmatrix}\frac{v_1}{\sqrt{\sum_{i=1}^nv_i^2}} \\ \vdots \\ \frac{v_n}{\sqrt{\sum_{i=1}^nv_i^2}} \end{pmatrix} =\vec{u} \\ \\ \Vert\vec{u}\Vert^2&amp;=\sum_{k=1}^n\left(\frac{v_k^2}{\sum_{i=1}^nv_i^2}\right) \end{align}  I think it is the last step that is either overcomplicated or wrong.
I am given the information that $ \vec{u} = \langle 1, 1/2 \rangle$ and $\vec{v} = \langle 2,3 \rangle$.   There are a few pieces I am asked to find, and these are the one I am having trouble with:   $\left\Vert \large \frac{\vec{u}}{|| \vec{u}||}\right\Vert$  The magnitude of vector u is $ \sqrt{5}/2$, of which I correctly calculated; the unit vector is then $ \langle \large \frac{2 \sqrt{5}}{5}, \frac{ \sqrt{5}}{5} \rangle$. However, when I go to find the magnitude of the unit vector, I get $ \sqrt \frac{3}{5}$, which is clearly not one. What did I do wrong?
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