If P, Q, R, S are the points (– 2, 3, 4), (– 4, 4, 6), (4, 3, 5), (0, 1, 2), prove by projection that PQ is perpendicular to RS.

The given points are P (– 2, 3, 4), Q (– 4, 4, 6), R (4, 3, 5) and S (0, 1, 2).
Direction ratios of RS are. 0 – 4, 1 – 3, 2 – 5 i.e., – 4, – 2, – 3
∴    direction-cosines of RS are
     
       projection of PQ on RS
                              

                                                                  
                                

∴ PQ is perpendicular to RS.
[∵ projection of a line perpendicular to it is zero]