Show that the line joining the origin to the point $(2,1,1)$ is perpendicular to the line determined by the points $(3,5, - 1),(4,3, - 1).$

.: The direction ratios of the line joining $O(0,0,0)$ and $A(2,1,1)$ are $< 2 - 0,1 - 0,1 - 0 >$i.e., $< 2,1,1 >$

The direction ratios of the line joining $B(3,5, - 1)$ and $C(4,3, - 1)$ are $< 4 - 3,3 - 5, - 1 + 1 >$ i.e., $< 1, - 2,0 >$

Now, $(2)(1) + (1)( - 2) + (1)(0) = 2 - 2 + 0 = 0$

Hence, OA is perpendicular to BC.