Question

What does orthogonal projection means?

Answer 1

Orthogonal projection:

Let $U$ be a subspace in $R^n$ and $x$ be a vector in $R^n$. The closest vector of $x$ in $U$ is $x_U$.

Then $x_U$ is known as the orthogonal projection of $x$ in $U$.

Orthogonal projection onto a line:

If $L=\mathrm{span}\left\{U\right\}$ is a line then

$x_L=\frac{u·u}{x·u}u$ and $x_{L\perp }=x-x_L$

for any vector $x$.