A) \[{{\cos }^{-1}}\left( \frac{1}{8} \right)\]
B) \[{{\cos }^{-1}}\left( \frac{1}{6} \right)\]
C) \[{{\cos }^{-1}}\left( \frac{1}{3} \right)\]
D) \[{{\cos }^{-1}}\left( \frac{1}{4} \right)\]
Correct Answer: B
Solution :
Given \[l+3m+5n=0......\]1 and \[5lm-2mn+6nl=0\].......2 Here \[l,m,n\] are directional cosines. From 1, \[l=-3m-5n\] Substituting equation 1 in equation 2 \[5(-3m-5n)m-2mn+6n(-3m-5n)=0\] \[15{{m}^{2}}+45mn+30{{n}^{2}}=0\]b \[\Rightarrow {{m}^{2}}+3mn+2{{n}^{2}}=0\] \[\Rightarrow {{m}^{2}}+2mn+mn+2{{n}^{2}}=0\] \[\Rightarrow (m+n)(m+2n)=0\] \[\therefore m=-n\] or \[m=-2n\] \[For\,\,\,m=-n;l=-2n\] And for \[m=-2n;l=n\] \[\therefore (l,m,n)=(-2n,-n,n)\] Or\[(l,m,n)=(n,-2n,n)\] \[\Rightarrow (\operatorname{l},m,n)=(-2,-1,1)\] Or \[\Rightarrow (l,m,n)=(1,-2,1)\] \[\cos (\theta )=\frac{A.B}{|A||B|},\theta \] is angle between the lines \[\Rightarrow \cos (\theta )=\frac{-2.1+(-1).(-2)+1.1}{\sqrt{6}.\sqrt{6}}\] \[\Rightarrow \cos (\theta )=\frac{1}{6}\] \[\therefore \theta ={{\cos }^{-1}}\left( \frac{1}{6} \right)\] Hence, correct option is ?B?
You need to login to perform this action.
You will be redirected in
3 sec
