Given two vectors $a$ and $b$ prove their distance is $2\sum_{i=1}^{n} (a_i + t_p h_i) h_i$

Question

Given vectors $a = (a_1, \ldots, a_n)$ and $b = (b_1, \ldots, b_n)$ prove that it exists $t_p \in \mathbb{R}$ such that the distance between $a$ and $b$ is $$ 2\sum_{i=1}^{n} (a_i + t_p h_i) h_i $$ where $h_i = b_i - a_i$.

I think it may be a limit case in Cauchy-Scharz inequality, but don't know how to prove it. Any help?

Answer 1

$$ t_p = \frac{|\vec{h}|-2\vec{a}\vec{h}}{2|\vec{h}|^2} $$