-- Diagram from Wiki
It is easier to imagine relation in three spatial vectors by their angles, than by their correlations. For standardized and s , , cosines of three angles of the triangular pyramid determinate the correlation matrix, thus, all statistics of the regressions and . Unexpected but imaginative results on the impact of introducing are --
1. Both s are nearly independent of . Togethor they predict almost perfectly ( and ).
2. Both s are almost perfectly correlated with . Togethor, one of the regressive coefficient is significantly negative (, and respectively).
3. Redundancy (Cohen, Cohen, West, & Aiken, 2003) increases to full and then decreases to zero and even negative (, and closes from to then to ).
Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences(3rd ed.) Mahwah, NJ: Lawrence Erlbaum Associates.