Question

The direction cosines $l,m,n$ of two lines are connected by the relations $l-5m+3m=0$ and $7l^2+5m^2-3n^2=0$ the angle between them is ${\cos }^{-1}?$

• A. $\frac{5}{\sqrt{84}}$
• B. $\frac{6}{\sqrt{84}}$
• C. $\frac{7}{\sqrt{84}}$
• D. $\frac{9}{\sqrt{84}}$

Answer 1

The correct option is C $\frac{7}{\sqrt{84}}$

In order to compute the values of $l,m,n$, we must first solve these equations

Given that

$l-5m+3n=0$ and $7l^2+5m^2-3n^2=0$

From equation 1, $l=5m-3n$

Substitute this value of $l$ in equation 2

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

On simplifying we get

$\Rightarrow 30(2m-n)(3m-2n)=0$

$\Rightarrow 2m=n;3m=2n$

Therefore $\Rightarrow \frac{m}{1}=\frac{n}{2}=\frac{5m-3n}{5-2.3}=\frac{l}{-1}=\frac{1}{\sqrt{6}}$

and

$\Rightarrow \frac{m}{2}=\frac{n}{3}=\frac{5m-3n}{5.2-3.3}=\frac{1}{\sqrt{14}}$

Hence the required dc's of the line are $\frac{-1}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{2}{\sqrt{6}}$ and $\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}$

We know that $cos\theta =l_1{l}_2+m_1{m}_2+n_1{n}_2$

$\Rightarrow cos\theta =\frac{-1}{\sqrt{84}}+\frac{2}{\sqrt{84}}+\frac{6}{\sqrt{84}}$

$\Rightarrow cos\theta =\frac{7}{\sqrt{84}}$

$\Rightarrow \theta =co{s}^{-1}\left(\frac{7}{\sqrt{84}}\right)$