If-a-variable-line-in-two-adjacent-positions-has-direction-cosines-l-m-n-and-l-l-m-m-n-n-show-that-the-small-angle-b-w-the-two-positions-is-given-by-2-l-2-m-2-n-2-

Question Number 75445 by vishalbhardwaj last updated on 11/Dec/19

If a variable line in two adjacent positions has direction cosines l,m, n and l+δl, m+δm, n+δn , show that the small angle δθ b/w the two positions is given by δθ^2 = δl^2 + δm^2 + δn^2 ??

$$\mathrm{If}\:\mathrm{a}\:\mathrm{variable}\:\mathrm{line}\:\mathrm{in}\:\mathrm{two}\:\mathrm{adjacent} \\ $$$$\mathrm{positions}\:\mathrm{has}\:\mathrm{direction}\:\mathrm{cosines} \\ $$$${l},{m},\:{n}\:\mathrm{and}\:{l}+\delta{l},\:{m}+\delta{m},\:{n}+\delta{n} \\ $$$$,\:\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{small}\:\mathrm{angle}\:\delta\theta\: \\ $$$$\mathrm{b}/\mathrm{w}\:\mathrm{the}\:\mathrm{two}\:\mathrm{positions}\:\mathrm{is}\:\mathrm{given}\:\mathrm{by} \\ $$$$\delta\theta^{\mathrm{2}} \:=\:\delta{l}^{\mathrm{2}} \:+\:\delta{m}^{\mathrm{2}} \:+\:\delta{n}^{\mathrm{2}} \:\:?? \\ $$

Answered by mr W last updated on 11/Dec/19

x_1 =l y_1 =m z_1 =n x_2 =l+δl y_2 =m+δm z_2 =n+δn (Δl)^2 =(Δx)^2 +(Δy)^2 +(Δz)^2 =(x_2 −x_1 )^2 +(y_2 −y_1 )^2 +(z_2 −z_2 )^2 =(δl)^2 +(δm)^2 +(δn)^2 ...(i) (Δl)^2 =1^2 +1^2 −2×1×1×cos (δθ) =2−2 [1−2 sin^2 (((δθ)/2))] =4 [sin ((δθ)/2)]^2 ≈4 [((δθ)/2)]^2 =(δθ)^2 ...(ii) ⇒(δθ)^2 =(δl)^2 +(δm)^2 +(δn)^2

$${x}_{\mathrm{1}} ={l} \\ $$$${y}_{\mathrm{1}} ={m} \\ $$$${z}_{\mathrm{1}} ={n} \\ $$$${x}_{\mathrm{2}} ={l}+\delta{l} \\ $$$${y}_{\mathrm{2}} ={m}+\delta{m} \\ $$$${z}_{\mathrm{2}} ={n}+\delta{n} \\ $$$$\left(\Delta{l}\right)^{\mathrm{2}} =\left(\Delta{x}\right)^{\mathrm{2}} +\left(\Delta{y}\right)^{\mathrm{2}} +\left(\Delta{z}\right)^{\mathrm{2}} \\ $$$$=\left({x}_{\mathrm{2}} −{x}_{\mathrm{1}} \right)^{\mathrm{2}} +\left({y}_{\mathrm{2}} −{y}_{\mathrm{1}} \right)^{\mathrm{2}} +\left({z}_{\mathrm{2}} −{z}_{\mathrm{2}} \right)^{\mathrm{2}} \\ $$$$=\left(\delta{l}\right)^{\mathrm{2}} +\left(\delta{m}\right)^{\mathrm{2}} +\left(\delta{n}\right)^{\mathrm{2}} \:\:\:…\left({i}\right) \\ $$$$ \\ $$$$\left(\Delta{l}\right)^{\mathrm{2}} =\mathrm{1}^{\mathrm{2}} +\mathrm{1}^{\mathrm{2}} −\mathrm{2}×\mathrm{1}×\mathrm{1}×\mathrm{cos}\:\left(\delta\theta\right) \\ $$$$=\mathrm{2}−\mathrm{2}\:\left[\mathrm{1}−\mathrm{2}\:\mathrm{sin}^{\mathrm{2}} \:\left(\frac{\delta\theta}{\mathrm{2}}\right)\right] \\ $$$$=\mathrm{4}\:\left[\mathrm{sin}\:\frac{\delta\theta}{\mathrm{2}}\right]^{\mathrm{2}} \\ $$$$\approx\mathrm{4}\:\left[\frac{\delta\theta}{\mathrm{2}}\right]^{\mathrm{2}} \\ $$$$=\left(\delta\theta\right)^{\mathrm{2}} \:\:\:…\left({ii}\right) \\ $$$$ \\ $$$$\Rightarrow\left(\delta\theta\right)^{\mathrm{2}} =\left(\delta{l}\right)^{\mathrm{2}} +\left(\delta{m}\right)^{\mathrm{2}} +\left(\delta{n}\right)^{\mathrm{2}} \\ $$

Commented by mr W last updated on 11/Dec/19

Commented by mr W last updated on 11/Dec/19

P_1 (x_1 ,y_1 ,z_1 ) P_2 (x_2 ,y_2 ,z_2 ) let OP_1 =OP_2 =1 x_1 =1 cos α=l y_1 =1 cos β=m z_1 =1 cos γ=n similarly x_2 =l+δl y_2 =m+δm z_2 =n+δn let Δl=P_1 P_2 Δl=(√((x_2 −x_1 )^2 +(y_2 −y_1 )^2 +(z_2 −z_1 )^2 )) =(√((δl)^2 +(δm)^2 +(δn)^2 )) ⇒(Δl)^2 =(δl)^2 +(δm)^2 +(δn)^2 on the other side, (P_1 P_2 )^2 =(OP_1 )^2 +(OP_2 )^2 −2(OP_1 )(OP_2 )cos ∠P_1 OP_2 i.e. (Δl)^2 =1^2 +1^2 −2×1×1 cos (δθ) =2−2 cos (δθ) =2−2(1−2 sin^2 ((δθ)/2)) =4 (sin ((δθ)/2))^2 ≈4 (((δθ)/2))^2 since δθ is small, sin ((δθ)/2)≈((δθ)/2) =(δθ)^2 ⇒(δθ)^2 =(δl)^2 +(δm)^2 +(δn)^2

$${P}_{\mathrm{1}} \left({x}_{\mathrm{1}} ,{y}_{\mathrm{1}} ,{z}_{\mathrm{1}} \right) \\ $$$${P}_{\mathrm{2}} \left({x}_{\mathrm{2}} ,{y}_{\mathrm{2}} ,{z}_{\mathrm{2}} \right) \\ $$$${let}\:{OP}_{\mathrm{1}} ={OP}_{\mathrm{2}} =\mathrm{1} \\ $$$${x}_{\mathrm{1}} =\mathrm{1}\:\mathrm{cos}\:\alpha={l} \\ $$$${y}_{\mathrm{1}} =\mathrm{1}\:\mathrm{cos}\:\beta={m} \\ $$$${z}_{\mathrm{1}} =\mathrm{1}\:\mathrm{cos}\:\gamma={n} \\ $$$${similarly} \\ $$$${x}_{\mathrm{2}} ={l}+\delta{l} \\ $$$${y}_{\mathrm{2}} ={m}+\delta{m} \\ $$$${z}_{\mathrm{2}} ={n}+\delta{n} \\ $$$${let}\:\Delta{l}={P}_{\mathrm{1}} {P}_{\mathrm{2}} \\ $$$$\Delta{l}=\sqrt{\left({x}_{\mathrm{2}} −{x}_{\mathrm{1}} \right)^{\mathrm{2}} +\left({y}_{\mathrm{2}} −{y}_{\mathrm{1}} \right)^{\mathrm{2}} +\left({z}_{\mathrm{2}} −{z}_{\mathrm{1}} \right)^{\mathrm{2}} } \\ $$$$=\sqrt{\left(\delta{l}\right)^{\mathrm{2}} +\left(\delta{m}\right)^{\mathrm{2}} +\left(\delta{n}\right)^{\mathrm{2}} } \\ $$$$\Rightarrow\left(\Delta{l}\right)^{\mathrm{2}} =\left(\delta{l}\right)^{\mathrm{2}} +\left(\delta{m}\right)^{\mathrm{2}} +\left(\delta{n}\right)^{\mathrm{2}} \\ $$$$ \\ $$$${on}\:{the}\:{other}\:{side}, \\ $$$$\left({P}_{\mathrm{1}} {P}_{\mathrm{2}} \right)^{\mathrm{2}} =\left({OP}_{\mathrm{1}} \right)^{\mathrm{2}} +\left({OP}_{\mathrm{2}} \right)^{\mathrm{2}} −\mathrm{2}\left({OP}_{\mathrm{1}} \right)\left({OP}_{\mathrm{2}} \right)\mathrm{cos}\:\angle{P}_{\mathrm{1}} {OP}_{\mathrm{2}} \\ $$$${i}.{e}. \\ $$$$\left(\Delta{l}\right)^{\mathrm{2}} =\mathrm{1}^{\mathrm{2}} +\mathrm{1}^{\mathrm{2}} −\mathrm{2}×\mathrm{1}×\mathrm{1}\:\mathrm{cos}\:\left(\delta\theta\right) \\ $$$$=\mathrm{2}−\mathrm{2}\:\mathrm{cos}\:\left(\delta\theta\right) \\ $$$$=\mathrm{2}−\mathrm{2}\left(\mathrm{1}−\mathrm{2}\:\mathrm{sin}^{\mathrm{2}} \:\frac{\delta\theta}{\mathrm{2}}\right) \\ $$$$=\mathrm{4}\:\left(\mathrm{sin}\:\frac{\delta\theta}{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$$\approx\mathrm{4}\:\left(\frac{\delta\theta}{\mathrm{2}}\right)^{\mathrm{2}} \:\:{since}\:\delta\theta\:{is}\:{small},\:\mathrm{sin}\:\frac{\delta\theta}{\mathrm{2}}\approx\frac{\delta\theta}{\mathrm{2}} \\ $$$$=\left(\delta\theta\right)^{\mathrm{2}} \\ $$$$ \\ $$$$\Rightarrow\left(\delta\theta\right)^{\mathrm{2}} =\left(\delta{l}\right)^{\mathrm{2}} +\left(\delta{m}\right)^{\mathrm{2}} +\left(\delta{n}\right)^{\mathrm{2}} \\ $$

Commented by vishalbhardwaj last updated on 11/Dec/19

$$\mathrm{sir}\:\mathrm{please}\:\mathrm{explain}\:\mathrm{in}\:\mathrm{detail} \\ $$

Commented by peter frank last updated on 11/Dec/19

$${thank}\:{you} \\ $$

Commented by vishalbhardwaj last updated on 11/Dec/19

but sir it is not given that the length of line is unit

$$\mathrm{but}\:\mathrm{sir}\:\mathrm{it}\:\mathrm{is}\:\mathrm{not}\:\mathrm{given}\:\mathrm{that}\:\mathrm{the}\: \\ $$$$\mathrm{length}\:\mathrm{of}\:\mathrm{line}\:\mathrm{is}\:\mathrm{unit} \\ $$

Commented by vishalbhardwaj last updated on 12/Dec/19

And sir this is a particular case because it is not given that lines are started from origin, Infact they can be started from any point in space except origin , Please give that general case to understand.

$$\mathrm{And}\:\mathrm{sir}\:\mathrm{this}\:\mathrm{is}\:\mathrm{a}\:\mathrm{particular}\:\mathrm{case} \\ $$$$\mathrm{because}\:\mathrm{it}\:\mathrm{is}\:\mathrm{not}\:\mathrm{given}\:\mathrm{that}\: \\ $$$$\mathrm{lines}\:\mathrm{are}\:\mathrm{started}\:\mathrm{from}\:\mathrm{origin},\: \\ $$$$\mathrm{Infact}\:\mathrm{they}\:\mathrm{can}\:\mathrm{be}\:\mathrm{started}\:\mathrm{from} \\ $$$$\mathrm{any}\:\mathrm{point}\:\mathrm{in}\:\mathrm{space}\:\mathrm{except}\:\mathrm{origin} \\ $$$$,\:\mathrm{Please}\:\mathrm{give}\:\mathrm{that}\:\mathrm{general}\:\mathrm{case} \\ $$$$\mathrm{to}\:\mathrm{understand}. \\ $$

Commented by mr W last updated on 12/Dec/19

since we only need to find the (small) change of the direction of the line, we can consider the change of a unit length segment of it and we can translate the segment such that one end of the segment is at the same point. the translation doesn′t affect the direction of the line.

$${since}\:{we}\:{only}\:{need}\:{to}\:{find}\:{the}\:\left({small}\right) \\ $$$${change}\:{of}\:{the}\:{direction}\:{of}\:{the}\:{line}, \\ $$$${we}\:{can}\:{consider}\:{the}\:{change}\:{of}\:{a} \\ $$$${unit}\:{length}\:{segment}\:{of}\:{it}\:{and}\:{we} \\ $$$${can}\:{translate}\:{the}\:{segment}\:{such} \\ $$$${that}\:{one}\:{end}\:{of}\:{the}\:{segment}\:{is}\:{at} \\ $$$${the}\:{same}\:{point}.\:{the}\:{translation} \\ $$$${doesn}'{t}\:{affect}\:{the}\:{direction}\:{of}\:{the} \\ $$$${line}. \\ $$

Commented by mr W last updated on 12/Dec/19

Commented by vishalbhardwaj last updated on 12/Dec/19

please sir if it is possible to you to use general case please post it

$$\mathrm{please}\:\mathrm{sir}\:\mathrm{if}\:\mathrm{it}\:\mathrm{is}\:\mathrm{possible}\:\mathrm{to}\:\mathrm{you} \\ $$$$\mathrm{to}\:\mathrm{use}\:\mathrm{general}\:\mathrm{case}\:\mathrm{please}\:\mathrm{post} \\ $$$$\mathrm{it} \\ $$

Commented by mr W last updated on 12/Dec/19