Q. 14.0( 75 Votes )

Show that the three lines with direction cosines are mutually perpendicular.

Answer :

We know that

If l1, m1, n1 and l2, m2, n2 are the direction cosines of two lines; and θ is the acute angle between the two lines; then cos θ = |l1l2 + m1m2 + n1n2|


If two lines are perpendicular, then the angle between the two is θ = 90°


For perpendicular lines, | l1l2 + m1m2 + n1n2 | = cos 90° = 0, i.e.


| l1l2 + m1m2 + n1n2 | = 0


So, in order to check if the three lines are mutually perpendicular, we compute | l1l2 + m1m2 + n1n2 | for all the pairs of the three lines.


Now let the direction cosines of L1, L2 and L3 be l1, m1, n1; l2, m2, n2 and l3, m3, n3.


First, consider




L1 L2 ……(i)


Next, consider




L2 L3 …(ii)


Now, consider




L1 L3 …(iii)


By (i), (ii) and (iii), we have


L1, L2 and L3 are mutually perpendicular.


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Questions Based on 3D GeometryQuestions Based on 3D GeometryQuestions Based on 3D Geometry26 mins
Questions Based on 3D Geometry | Check YourselfQuestions Based on 3D Geometry | Check YourselfQuestions Based on 3D Geometry | Check Yourself40 mins
Revision of straight lines important formulas in one shotRevision of straight lines important formulas in one shotRevision of straight lines important formulas in one shot59 mins
Concept Builder Questions of straight Lines (Quiz Session)Concept Builder Questions of straight Lines (Quiz Session)Concept Builder Questions of straight Lines (Quiz Session)55 mins
Reminding 11th - Revision of Circles important formulas in 50 MinutesReminding 11th - Revision of Circles important formulas in 50 MinutesReminding 11th - Revision of Circles important formulas in 50 Minutes56 mins
Conic Section Part 1Conic Section Part 1Conic Section Part 164 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses