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

Explanation

We have `l,m,n` and `l+deltl,m+deltam,n+deltan` as direction cosines of a variable line in two different positions.
`:. l^(2)+m^(2)+n^(2)=1`……………..i
and `(l+deltal)^(2)+(m+deltam)^(2)+(n+deltan)^(2)=1`……………ii
`impliesl^(2)+m^(2)+n^(2)+deltal^(2)+deltam^(2)+deltan^(2)+2(l delta l+m delta m+n delta n)=1`
`implies delta l^(2)+deltam^(2)+deltan^(2)=-2(l delta l+mdeltam+ndelta n) [ :' l^(2)+m^(2)+n^(2)=1]`
`impliesl delta l +m delta m +n delta n=(-1)/2 (delta l^(2)+deltam^(2)+deltan^(2))`.........iii
Now `veca` and `vecb` are unit vectors along a line with direction cosines `l,m,n` and `(l+deltal),(m+deltam),(n+delta n`) respectively
`:. veca=lhati+mhatj+nhatk` and `vecb=(l+deltal)hati+(m+deltam)hatj+(n+deltan)hatk`
`implies cos delta theta=(veca.vecb)/(|veca||vecb|)=veca.vecb [ :' |hata|=|hatb|=1]`
`impliescos delta theta=l(l+deltal)+m(m+deltam)+n(n+deltan)`
`=(l^(2)+m^(2)+n^(2))+(l delta l+m delta m+n deltan)`
`=1 - 1/2 (delta l^(2)+delta m^(2)+delta n^(2))` [using eq. iii
`implies 2(1-cos delta theta)=(delta l^(2)+delta m^(2)+delta n^(2))`
`implies 2.2 "sin"^(2)(delta theta)/2=delta l^(2)+delta m^(2)+delta n^(2)[ :'1-cos theta =2 "sin"^(2)(theta)/2]`
`implies 4((delta theta)/2)^(2)=delta l^(2)+delta m^(2)+deltan^(2)` [ since `(delta theta)/2` is small then `"sin"(delta theta)/2=(delta theta)/2`]
`:. delta theta^(2)=delta l^(2)+delta m^(2)+delta n^(2)`