Let $L$ be the line in ${\mathbb R} ^3$ that consists of all scalar multiples of the vector $\left[\begin{array}{c} 1\cr 2\cr 2 \end{array}\right].$ Find the orthogonal projection of the vector $\vec{x} = \left[\begin{array}{c} 2\cr 7\cr 4 \end{array}\right]$ onto $L$.
${\rm proj}_L \vec{x} =$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$