Could principle component decomposed coordinates value be correlated to each other?
PCs are calculated based on eigenvectors (e1 nad e2 in your case) of correlation or covariance matrix. Eigenvectors are orthogonal, so PCs are uncorrelated. If you would like to check the proof, you can find it on page 432 in Applied Multivariate Statistical Analysis 6th edition written by Richard A. Johnson and Dean W. Wichern.