Question Concerning Vectors

Question

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?

Answer 1

Observe that $\left\Vert\langle \large \frac{2 \sqrt{5}}{5}, \frac{ \sqrt{5}}{5} \rangle \right\Vert^2 = \displaystyle \bigl(\frac{2\sqrt{5}}{5}\bigr)^2+\bigl(\frac{\sqrt{5}}{5}\bigr)^2=\frac{20}{25}+\frac{5}{25}=1$.

Also note that $\displaystyle \left\Vert \frac{\vec{u}}{\Vert \vec{u} \Vert} \right \Vert=\left| \frac{1}{\vert \vert \vec{u} \vert\vert} \right| \cdot \vert \vert \vec{u} \vert |=\frac{1}{\vert \vert \vec{u} \vert\vert}\cdot \vert \vert \vec{u} \vert \vert=1$.