I have a problem with the formula of orthonormal projection.
I know the orthogonal projection of a vector $v$ to one dimension subspace $U$ with basis $u$ is $$ Pro^{v}_{U}=\dfrac{\langle v,u \rangle }{\langle u,u \rangle }u$$
but from the other hand the length of projection vector is inner product of $u$ and $v$, $\langle u,v \rangle .$
Now my problem is that if I want to find projection length from the first formula I have:
$$\langle Pro^{v}_{U},Pro^{v}_{U} \rangle =\langle \dfrac{\langle v,u \rangle }{\langle u,u \rangle }u,\dfrac{\langle v,u \rangle }{\langle u,u \rangle }u \rangle $$
which it is not equal to inner product of $u$ and $v$.
Where am I making mistake?
These 2 are not equal for example: $v=(2,3)$ , $u=(1,2)$
$$Pro^{v}_{U}=(\frac{8}{5},\frac{16}{5}),$$ $$\langle Pro^{v}_{U},Pro^{v}_{U} \rangle =\frac{320}{25},$$
but $ \langle u,v \rangle =8.$
The lengh of projection vector is equal to $\langle u, v \rangle$ for $|u|=1$ and in this case
$$\mbox{Pro}^{v}_{U}= \langle v, u \rangle u$$
and
$$\langle v,u \rangle = \sqrt{\langle \mbox{Pro}^{v}_{U}, \mbox{Pro}^{v}_{U}\rangle} = \sqrt{\langle \langle v,u \rangle u,\langle v,u \rangle u \rangle}$$