If l1, m1, n1 and l2, m2, n2 are the direction cosines of two mutually perpendicular lines,

It is given that l1, m1, n1 and l2, m2, n2 are the direction cosines of two mutually perpendicular lines. Therefore,

$l_{1} l_{2}+m_{1} m_{2}+n_{1} n_{2}=0$                         ...(1)

$l_{1}^{2}+m_{1}^{2}+n_{1}^{2}=1$                                   ...(2)

$l_{2}^{2}+m_{2}^{2}+n_{2}^{2}=1$                                   ...(3)

Let l, m, n be the direction cosines of the line which is perpendicular to the line with direction cosines l1, m1, n1 and l2, m2, n2.

$\therefore l l_{1}+m m_{1}+n n_{1}=0$

$l l_{2}+m m_{2}+n n_{2}=0$

$\therefore \frac{l}{m_{1} n_{2}-m_{2} n_{1}}=\frac{m}{n_{1} l_{2}-n_{2} l_{1}}=\frac{n}{l_{1} m_{2}-l_{2} m_{l}}$

$\Rightarrow \frac{l^{2}}{\left(m_{1} n_{2}-m_{2} n_{1}\right)^{2}}=\frac{m^{2}}{\left(n_{1} l_{2}-n_{2} l_{1}\right)^{2}}=\frac{n^{2}}{\left(l_{1} m_{2}-l_{2} m_{l}\right)^{2}}$

$\Rightarrow \frac{l^{2}}{\left(m_{1} n_{2}-m_{2} n_{1}\right)^{2}}=\frac{m^{2}}{\left(n_{1} l_{2}-n_{2} l_{1}\right)^{2}}=\frac{n^{2}}{\left(l_{1} m_{2}-l_{2} m_{2}\right)^{2}}$

$=\frac{l^{2}+m^{2}+n^{2}}{\left(m_{1} n_{2}-m_{2} n_{1}\right)^{2}+\left(n_{1} l_{2}-n_{2} l_{1}\right)^{2}+\left(l_{1} m_{2}-l_{2} m_{l}\right)^{2}}$     $\ldots(4)$

l, m, n are the direction cosines of the line.

$\therefore /^{2}+m^{2}+n^{2}=1 \ldots$ (5)

It is known that,

$\left(l_{1}^{2}+m_{1}^{2}+n_{1}^{2}\right)\left(l_{2}^{2}+m_{2}^{2}+n_{2}^{2}\right)-\left(l_{1} l_{2}+m_{1} m_{2}+n_{1} n_{2}\right)^{2}$

$\therefore\left(m_{1} n_{2}-m_{2} n_{1}\right)^{2}+\left(n_{1} l_{2}-n_{2} l_{1}\right)^{2}+\left(l_{1} m_{2}-l_{2} m_{1}\right)^{2}=1$                     ...(6)

Substituting the values from equations (5) and (6) in equation (4), we obtain

$\frac{l^{2}}{\left(m_{1} n_{2}-m_{2} n_{1}\right)^{2}}=\frac{m^{2}}{\left(n_{2} l_{2}-n_{2} l_{1}\right)^{2}}=\frac{n^{2}}{\left(l_{1} m_{2}-l_{2} m_{1}\right)^{2}}=1$

$\Rightarrow l=m_{1} n_{2}-m_{2} n_{1}, m=n_{1} l_{2}-n_{2} l_{1}, n=l_{1} m_{2}-l_{2} m_{1}$

Thus, the direction cosines of the required line are $m_{1} n_{2}-m_{2} n_{1}, n_{1} l_{2}-n_{2} l_{1}$, and $l_{1} m_{2}-l_{2} m_{1}$.