Question

The dc's $(l,m,n)$ of two lines are connected between the relation $l+m+n=0,lm=0$, then the angle between the lines is

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{6}$

Answer 1

The correct option is A $\frac{\pi }{3}$

We know that,

$\cos \theta =\frac{\overrightarrow{b_1.}\overrightarrow{b_2}}{\overrightarrow{\left|b_1\right|}\left|{\overrightarrow{b}}_2\right|}$ ...... (1)

Given that,

$l+m+n=0$

$l+m=-n$

$-(l+m)=n$

And, $lm=0$

So,

Either $l=0$ or $m=0$

When, $l=0$, then

$m=-n$

And,

$(l,m,n)=(0,1,-1)$

When, $m=0$, then

$l=-n$

And,

$(I,m,n)=(1,0,-1)$

Calculate $b_1\cdot b_2$.

$b_1.b_2=(0,1,-1).(1,0,-1)$

$=0+0+1$

$=1$

Therefore,

$\left|b_1\right|=\sqrt{0^2+1^2+{(-1)}^2}=\sqrt{2}$

$\left|b_2\right|=\sqrt{0^2+1^2+{(-1)}^2}=\sqrt{2}$

Now, substitute the values in equation (1).

$\cos \theta =\frac{\overrightarrow{b_1}.\overrightarrow{b_2}}{\left|{\overrightarrow{b}}_1\right|\left|\overrightarrow{b_2}\right|}$

$\Rightarrow \cos \theta =\frac{1}{\sqrt{2}.\sqrt{2}}=\frac{1}{2}$

$\Rightarrow \theta =\frac{\pi }{3}$

Hence, the angle between the lines is $\frac{\pi }{3}$.