A vector quantity is independent of the choice of coordinate system. The vectors \vec{P}, \vec{Q}, \vec{R} themselves do not change when the coordinate axes are changed; only their components change.

\vec{P} + \vec{Q} + \vec{R} is a vector sum, so it is independent of the coordinate system.

\(\left(P_{x} + Q_{x} + R_{x}\right) \hat{i}\) depends on the x -axis chosen.

P_x \hat{i} + Q_y \hat{j} + R_z \hat{k} depends on the orientation of the coordinate axes.

Therefore, and are coordinate-system dependent.