Say I have 2 points $P_1 (x_1, y_1)$ and $P_2 (x_2, y_2)$, such that $x_1 > x_2, y_1 > y_2$.
When $P_1$ and $P_2$ undergoes a orthogonal projection onto a vector $V(v_x, v_y)$, to obtain, $P_1' (x'_1, y'_1)$ and $P_2' (x'_2, y'_2)$.
In this case, can I say that for all cases, $x'_1 > x'_2, y'_1 > y'_2$?
No. Consider projecting onto the vector $\langle 1, 0 \rangle$. Then the $y$-coordinates of $P'_1$ and $P'_2$ will both be $0$.
Strict inequality doesn't hold either. Let $P_1 = (2,1), P_2 = (0,0)$ and let the vector be $\langle 1, -1 \rangle$.