Category: Vector Calculus

Question Number 5115 by Yozzii last updated on 14/Apr/16 $$Giventhespacecurve\mathit{r}=\mathit{r}\left(t\right),show$$$$thatitstorsion\tau isgivenby$$$$\tau =\frac{\stackrel{.}{\mathit{r}}\bullet \stackrel{..}{\mathit{r}}\times \stackrel{\dots }{\mathit{r}}}{\mid \stackrel{.}{\mathit{r}}\times \stackrel{..}{\mathit{r}}{\mid }^2}.Itmayhelptoknowthatits$$$${curvature}\:{is}\:{numerically}\:{given}\:{by}\:\kappa=\frac{\mid\overset{.} {\boldsymbol{{r}}}×\overset{..} {\boldsymbol{{r}}}\mid}{\mid\overset{.} {\boldsymbol{{r}}}\mid^{\mathrm{3}}…

Question Number 4772 by Yozzii last updated on 07/Mar/16 $$Letz={Ax}^2+Bxy+{Cy}^2.Findconditions$$$$ontheconstantsA,B,Cthatensure$$$$thatthepoint(0,0,0)isa$$$$\left(i\right)localminimum,$$$$\left(ii\right)localmaximum,$$$$\left(ii\right)saddlepoint.$$$$ \…

Question Number 135630 by rexford last updated on 14/Mar/21 Answered by mr W last updated on 14/Mar/21 Commented by mr W last updated on 14/Mar/21…

Question Number 134949 by rexford last updated on 08/Mar/21 Answered by bobhans last updated on 30/Jan/22 $$\overrightarrow{\mathrm{b}}=2\overrightarrow{\mathrm{c}}+\lambda \overrightarrow{\mathrm{a}};\mid \overrightarrow{\mathrm{b}}\mid =\mid 2\overrightarrow{\mathrm{c}}+\lambda \overrightarrow{\mathrm{a}}\mid$$$$\Rightarrow\:\mathrm{4}\:=\:\sqrt{\mid\mathrm{2}\overset{\rightarrow}…

Question Number 133937 by rexford last updated on 25/Feb/21 Answered by EDWIN88 last updated on 25/Feb/21 $$\mathit{A}\mathit{B}=\hat{\mathbf{i}}+6\hat{\mathbf{j}},\mathrm{let}\mathrm{vector}\mathbf{u}=\mathbf{AC}\mathrm{where}\mathrm{C}(\mathrm{x},\mathrm{y})$$$$\mathbf{u}=(\mathrm{x}-1,\mathrm{y}+1)=(\mathrm{x}-1)\hat{\mathbf{i}}+(\mathrm{y}+1)\hat{\mathbf{j}}$$$$\Rightarrow\:\boldsymbol{\mathrm{u}}.\boldsymbol{\mathrm{AB}}\:=\mathrm{0}\:\Rightarrow\mathrm{x}−\mathrm{1}+\mathrm{6y}+\mathrm{6}\:=\:\mathrm{0}…