Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • The direction cosines of a line equally inclined to three mutually perpendicular lines having D.C.’s as <img alt="\ l_{1}, m_{1}, n_{1} ; \ l_{2}, m_{2}, n_{2} ; \ l_{3}, m_{3}, n_{3}" src="h

The direction cosines of a line equally inclined to three mutually perpendicular lines having D.C.’s as \ l_{1}, m_{1}, n_{1} ; \ l_{2}, m_{2}, n_{2} ; \ l_{3}, m_{3}, n_{3} are

Option: 1

\ l_{1}+\ l_{2}+\ l_{3},m_{1}+m_{2}+m_{3},n_{1}+n_{2}+n_{3}
 


Option: 2

\frac{\ l_{1}+\ l_{2}+\ l_{3}}{\sqrt{3}},\frac{m_{1}+m_{2}+m_{3}}{\sqrt{3}},\frac{n_{1}+n_{2}+n_{3}}{\sqrt{3}}

 


Option: 3

\frac{\ l_{1}+\ l_{2}+\ l_{3}}{3},\frac{m_{1}+m_{2}+m_{3}}{3},\frac{n_{1}+n_{2}+n_{3}}{3}


Option: 4

none of these


Answers (1)

best_answer

Ange between two lines in terms of direction cosines and direction ratios -

(i)    if two lines are parallel then

        l_{1}=l_{2}, m_{1}=m_{2}, n_{1}=n_{2} or

        \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}

(ii)     if two lines are perpendicular then

        l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2}= 0 or1

        a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}= 0

 

-

 


Let \overrightarrow{OA}=l_{1}\widehat{i}+m_{1}\widehat{j}+n_{1}\widehat{k},    \overrightarrow{OB}=l_{2}\widehat{i}+m_{2}\widehat{j}+n_{2}\widehat{k},  and  \overrightarrow{OC}=l_{3}\widehat{i}+m_{3}\widehat{j}+n_{3}\widehat{k}  be mutually perpendicular vectors .

Let \overrightarrow{OP}=l\widehat{i}+m\widehat{j}+n\widehat{k} be equally inclined to \overrightarrow{OA},\overrightarrow{OB}\:\:and\:\:\overrightarrow{OC}

Then

\overrightarrow{OP}=\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC} =\sum l_{1}\widehat{i}+\sum m_{1}\widehat{j}+\sum n_{1}\widehat{k}

\left | \overrightarrow{OP} \right |^{2}=\left ( \sum l_{1} \right )^{2}+\left ( \sum m_{1} \right )^{2}+\left ( \sum n_{1} \right )^{2}

=3+2\sum \left ( l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2} \right )=3

\therefore \left | \overrightarrow{OP} \right |=\sqrt{3}

\therefore \widehat{OP}=\frac{l_{1}+l_{2}+l_{3}}{\sqrt{3}}\widehat{i}+\frac{m_{1}+m_{2}+m_{3}}{\sqrt{3}}\widehat{j}+\frac{n_{1}+n_{2}+n_{3}}{\sqrt{3}}\widehat{k}

 

Posted by

qnaprep

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

NTA Exams 2025 Live: NEET UG date, registration form soon at neet.nta.nic.in; UGC NET revised schedule out
anam-khan

Is it difficult to get into NIT Durgapur? JEE Main cut-offs for BTech programmes
anam-khan

JEE Main Deleted Syllabus 2025: Topics to skip this year; why no optional questions?

Latest Question