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

If $I_{1}, m_{1}, n_{1}$ and $I_{2}, m_{2}, n_{2}$ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are $m_{1} n_{2}$ $m_{2} n_{1}, n_{1} l_{2}-n_{2} l_{1}, l_{1} m_{2}-l_{2} m_{1}$.

Solution:

It is given that l1m1n1 and l2m2n2 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 lmn be the direction cosines of the line which is perpendicular to the line with direction cosines l1m1n1 and l2m2n2.

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

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

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

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

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

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

lmn 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 $l_{1} m_{2}-l_{2} m_{1}$.