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if (l1,m1,n1),(l2,m2,n2),(l3,m3,n3) are the direction cosines of three mutual perpendicular lines,show that the line whose direction ratios area l1+l2+l3,m1+m2+m3,n1+n2+n3 make equal angles with them

if (l1,m1,n1),(l2,m2,n2),(l3,m3,n3) are the direction cosines of three mutual perpendicular lines,show that the line whose direction ratios area l1+l2+l3,m1+m2+m3,n1+n2+n3 make equal angles with them

Grade:12th pass

1 Answers

Vikas TU
14149 Points
4 years ago
Because the first three lines are mutually perpendicular, dot products of pairs of lines are zero. 
l1*l2+m1*m2+n1*n2 = 0 
l2*l3+m2*m3+n2*n3 = 0 
l1*l3+m1*m3+n1*n3 = 0 

Finally, using the above equations, and another dot product, it is shown that the cosine of the angle between the fourth vector and any of the first three vectors is the same.
Hence, it makes equal angles with them.

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