Question Number 5115 by Yozzii last updated on 14/Apr/16
![Given the space curve r=r(t), show that its torsion τ is given by τ=((r^. •r^(..) ×r^(...) )/(∣r^. ×r^(..) ∣^2 )). It may help to know that its curvature is numerically given by κ=((∣r^. ×r^(..) ∣)/(∣r^. ∣^3 )). r^. is differentiation of r once with respect to t.](https://www.tinkutara.com/question/Q5115.png)
$${Given}\:{the}\:{space}\:{curve}\:\boldsymbol{{r}}=\boldsymbol{{r}}\left({t}\right),\:{show} \\ $$$${that}\:{its}\:{torsion}\:\tau\:{is}\:{given}\:{by} \\ $$$$\tau=\frac{\overset{.} {\boldsymbol{{r}}}\bullet\overset{..} {\boldsymbol{{r}}}×\overset{…} {\boldsymbol{{r}}}}{\mid\overset{.} {\boldsymbol{{r}}}×\overset{..} {\boldsymbol{{r}}}\mid^{\mathrm{2}} }.\:{It}\:{may}\:{help}\:{to}\:{know}\:{that}\:{its} \\ $$$${curvature}\:{is}\:{numerically}\:{given}\:{by}\:\kappa=\frac{\mid\overset{.} {\boldsymbol{{r}}}×\overset{..} {\boldsymbol{{r}}}\mid}{\mid\overset{.} {\boldsymbol{{r}}}\mid^{\mathrm{3}} }. \\ $$$$\overset{.} {\boldsymbol{{r}}}\:{is}\:{differentiation}\:{of}\:\boldsymbol{{r}}\:{once}\:{with} \\ $$$${respect}\:{to}\:{t}. \\ $$