Question

The direction cosines of the two lines are determine by the relation $\ell -5m+3n=0$ and $7\ell +5m^2-3n^2=0$ find them ?

Answer 1

In order-to compute the values of l, m and n from the given relations, we shall first solve these equations

We have

$l-5m+3n=0$

$\Rightarrow l=5m-3n$

Substituting this value in the second equation we have

$7(5m-3n{)}^2+5m^2-3n^2=0$

Hence, $30(2m-n)(3m-2n)=0$ i.e. $2m=nand3m=2n$.

Therefore,

$\begin{array}{c}m/1=n/1=\left(5m-3n\right)/5-2.3=l/\left(-1\right)=1/\sqrt{6}\\ andm/2=n/3=\left(5m-3n\right)/\left(5.2-3.3\right)=l/1=1/\sqrt{14}\end{array}$

Hence, the required dc's of the line are

$-1/\sqrt{6},1/\sqrt{6},2/\sqrt{6},and1/\sqrt{14},2/\sqrt{14},3/\sqrt{14}$