If a variable line in two adjacent positions has direction cosines `l,m,n` and `l+deltal,m+deltam,n+delta n`, then show that the small angle `delta theta` between the two positions is given by `delta theta^(2)=delta l^(2)+deltam^(2)+deltan^(2)`.

Explanation

Since l, m and n, and `(l+deltam),(m+deltam),(n+deltan)` are the direction cosines, we have
`l^(2)+m^(2)+n^(2)=1" ".....(i)`
`(l^(2)+deltal)^(2)+(m+deltam)^(2)+(n+deltan)^(2)=`
or `l^(2)+m^(2)+n^(2)+2ldeltal+2mdeltam+2ndeltan+(deltal)^(2)+(deltam)^(2)+(deltan)^(2)=1`
or `2(ldelta^(2)+mdeltam+ndeltan)`
`=-({deltal)^(2)+(deltam)^(2)+(deltam)^(2)}" "....(ii)`
Now it is given that `deltatheta` is the angle between two adjacent positions of the line. Therefore
`cosdeltatheta=l(l+deltal)+m(m+deltam)+n(n+deltan)" "(iii)`
Now `cosdeltatheta=1-((deltatheta)^(2))/(2!)+((deltatheta)^(2))/(4!)-....` If `deltatheta` is small, then `cosdeltatheta=1-((deltatheta)^(2))/(2)`
Then from (iii), we have
`1-((deltatheta)^(2))/(2)=(l^(2)+m^(2)+n^(2))`
`
+(ldeltal+mdeltam+ndeltan)`
or `1-((deltatheta)^(2))/(2)=1-(1)/(2){(deltal)^(2)+(deltam)^(2)+(deltan)^(2)}`
[using (i) and (ii)]
or `(deltatheta)^(2)=(deltal)^(2)+(deltam)^(2)+(deltan)^(2)`