Consider the following design matrix, representing four sample points $x_{i}in{R}^2.$

$x=\left[\begin{array}{cc}4& 1\\ 2& 3\\ 5& 4\\ 1& 0\end{array}\right].$

We want to represent the data in only one dimension, so we turn to principal components analysis $($PCA$).$

$(1)[5$ pts$]$ Compute the unit$-$length principal component directions of $x,$ and state which one the PCA algorithm would choose if you request just one principal component. Please provide an exact answer, without approximation. $($You will need to use the square root symbol.$)$ Show your work!

$(2)[5$ pts$]$ The plot below depicts the sample points from $x.$ We want a one$-$dimensional representation of the data, so draw the principal component direction $($as a line$)$ and the projections of all four sample points onto the principal direction.

Label each projected point with its principal coordinate value $($where the origin's principal coordinate is zero$).$ Give the principal coordinate values exactly.

$(3)[4$ pts$]$ The plot below depicts the sample points from $x$ rotated $30$ degrees counterclockwise about the origin. As in part $($b$),$ identify the principal component direction that the PCA algorithm would choose and draw it $($as a line$)$ on the plot. Also draw the projections of the rotated points onto the principal direction.

Label each projected point with the exact value of its principal coordinate.