Relative to a fixed origin $O$ , the path of a high-precision industrial laser cutter $L$ is described by the vector equation

\mathbf{r} = \begin{pmatrix} 3 \\ -2 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix}

where $λ\lambda$ is a scalar parameter.

Given that $OA→\overrightarrow{OA}$ is a unit vector parallel to the path $L$ ,

find $OA→\overrightarrow{OA}$ .

The point $X$ lies on the path $L$ such that it is the closest point on the path to the origin $O$ .

Find the coordinates of $X$ .

Points $O$ , $X$ , and $A$ form a triangle $O X A$ .

Determine the exact area of triangle $O X A$ .