Solution:

Let’s consider $O A$ be the line joining the origin $(0,0,0)$ and the point $A(2,1,1)$.
And let $B C$ be the line joining the points $B(3,5,-1)$ and $C(4,3,-1)$
Therefore the direction ratios of $0 A=\left(a_{1}, b_{1}, c_{1}\right) \equiv[(2-0),(1-0),(1-0)] \equiv(2,1,1)$
And the direction ratios of $\mathrm{BC}=\left(\mathrm{a}_{2}, \mathrm{~b}_{2}, \mathrm{c}_{2}\right) \equiv[(4-3),(3-5),(-1+1)] \equiv(1,-2,0)$
It is given that,
$\mathrm{OA}$ is $\perp$ to $\mathrm{BC}$
We now have to prove that:
$a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0$
Let’s consider $L H S: a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}$
$\begin{array}{l}
a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=2 \times 1+1 \times(-2)+1 \times 0 \\
=2-2 \\
=0
\end{array}$
It is known to us that R.H.S is 0
So $\mathrm{LHS}=\mathrm{RHS}$
$\therefore \mathrm{OA}$ is $\perp$ to $\mathrm{BC}$
As a result, hence proved.