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, thenδ$$l2+$$δ$$m2+$$δ$$n2=$$

Question

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, thenδ$$l2+$$δ$$m2+$$δ$$n2=$$

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If θ is the acute angle between two lines having direction ratios $$a1$$,$$b1$$,$$c1and$$ $$a2$$,$$b2$$,$$c2$$ then $$cos$$θ$$=a1a2+b1b2+c1c2a12+b12+c12$$⋅$$a22+b22+c22It$$ implies that $$cos$$δθ$$=ll+$$δ$$l+mm+$$δ$$m+nn+$$δ$$n=1+l$$δ$$l+m$$δ$$m+n$$δ$$nAndl+$$δ$$l2+m+$$δ$$m2+n+$$δ$$n2=1l2+m2+n2+2l$$δ$$l+2m$$δ$$m+2n$$δ$$n+$$δ$$l2+$$δ$$m2+$$δ$$n2=12l$$δ$$l+m$$δ$$m+n$$δ$$n+$$δ$$l2+$$δ$$m2+$$δ$$n2=0It$$ implies that δ$$l2+$$δ$$m2+$$δ$$n2=$$−$$2l$$δ$$l+m$$δ$$m+n$$δ$$n=21$$−$$cos$$δθ$$=4sin2$$δθ$$2$$≅$$4$$δθ$$22$$ ∵$$sin$$θ≅θ for small θ$$=$$δθ$$2Therefore$$,δ$$l2+$$δ$$m2+$$δ$$n2=$$δθ$$2$$

If a variable line in two adjacent positions has direction cosines $(l, m, n) a n d (l + δ l, m + δ m, n + δ n)$ and $δ θ$ is the angle between the two positions, then ${(δ l)}^{2} + {(δ m)}^{2} + {(δ n)}^{2} =$

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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=

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If a variable line in two adjacent positions has direction cosines $(l, m, n) a n d (l + δ l, m + δ m, n + δ n)$ and $δ θ$ is the angle between the two positions, then ${(δ l)}^{2} + {(δ m)}^{2} + {(δ n)}^{2} =$