If the direction cosines of a variable line in two adjacent points be `l, M, n and l+deltal,m+deltam+n+deltan` the small angle `deltatheta`as between the two positions is given by

If the direction cosines of a variable line in two adjacent points be

1.	If the direction cosines of a variable line in two adjacent points be `l, M, n and l+deltal,m+deltam+n+deltan` the small angle `deltatheta`as between the two positions is given by
Answer» Clearly, we have `l^(2)+m^(2)+n^(2)=1" "...(i)` and`(l+deltal)^(2)+(m+deltam)^(2)+(n+deltan)^(2)=1" "...(ii)` Substracting (i) from (ii), we get `(l+deltal)^(2)+(m+deltam)^(2)+(n+deltan)^(2)-(l^(2)+m^(2)+n^(2))=0` `rArr (deltal)^(2)+(deltam)^(2)+(deltan)^(2)=-2(ldeltal+mdeltam+ndeltan)" "...(iii)` `therefore cosdeltatheta=l(l+deltal)+m(m+deltam)+n(n+deltan)` `=(l^(2)+m^(2)+n^(2))+(ldeltal+mdeltam+ndeltan)` `=1-(1)/(2)[(deltal)^(2)+(deltam)^(2)+(deltan)^(2)]` [using (i) and (iii)]. `therefore(deltal)^(2)+(deltam)^(2)+(deltan)^(2)=2(1-cosdeltatheta)` `=4sin^(2)"(deltatheta)/(2)=4((deltatheta)/(2))^(2)` `[therefore (deltatheta)/(2) " being small, " sin""(deltatheta)/(2)=(deltatheta)/(2)]` `(deltatheta)^(2)`. Hence, `(deltatheta)^(2)=(deltal)^(2)+(deltam)^(2)+(deltan)^(2)`.