Orthogonalisation Say we have a number of vectors $h_1=(2,0)$, $h_2=(1,3)$ which are not orthogonal to each other.
There is a way to make them orthogonal with the help of this orthogonalisation formula: 
Here I will use $\langle e_1,e_1\rangle$ to show the dot product of 2 vectors.
$\begin{align}
e_1 & = h_1 & = (2,0)
\\
e_2 & = h_2 - \frac{\langle h_2,e_1\rangle}{\langle e_1,e_1\rangle} e_1.
\end{align}$
$e_2$ will be orthogonal to $e_1$.   So we can conclude that the  vector $h_2$ is corrected with the help of vector $e_1$ and the quotient ($\frac{\langle h_2,e_1\rangle}{\langle e_1,e_1\rangle}$).
So my question is:   Where does this quotient come from ($\frac{\langle h_2,e_1\rangle}{\langle e_1,e_1\rangle}$) ?

$\dfrac{\langle\, h_2,e_1\,\rangle}{\langle\, e_1,e_1\,\rangle}e_1$ is the orthogonal projection of $h_2$ onto the line directed by $e_1$. When you substract from a vector its orthogonal projection onto the line directed by $e_1$, you get a vector orthogonal to $e_1$ (make a sketch!).
