If a variable line in two adjacent positions has direction cosines l,m,n  and  l+δl,m+δm,n+δn and δθ is the angle between the two positions, thenδl2+δm2+δn2=

If θ is the acute angle between two lines having direction ratios a1,b1,c1and a2,b2,c2 then cosθ=a1a2+b1b2+c1c2a12+b12+c12⋅a22+b22+c22It implies that cosδθ=ll+δl+mm+δm+nn+δn=1+lδl+mδm+nδnAndl+δl2+m+δm2+n+δn2=1l2+m2+n2+2lδl+2mδm+2nδn+δl2+δm2+δn2=12lδl+mδm+nδn+δl2+δm2+δn2=0It implies that δl2+δm2+δn2=−2lδl+mδm+nδn=21−cosδθ=4sin2δθ2≅4δθ22 ∵sinθ≅θ for small θ=δθ2Therefore,δl2+δm2+δn2=δθ2