diff --git a/labs-set-2/template.pdf b/labs-set-2/template.pdf index 099c6ae..768ff36 100644 Binary files a/labs-set-2/template.pdf and b/labs-set-2/template.pdf differ diff --git a/labs-set-2/template.tex b/labs-set-2/template.tex index 3ede392..dce23e2 100644 --- a/labs-set-2/template.tex +++ b/labs-set-2/template.tex @@ -598,5 +598,45 @@ This corresponds with answer choice B. \end{align*} This corresponds with answer choice A. } +\qs{}{ + \begin{align*} + \vec{r}(t) &= \langle 2t-7,\; t^2-2,\; 7\rangle \\ + \vec{r}^{\prime}(t) &= \langle 2,\; 2t,\; 0\rangle \\ + \vec{r}^{\prime\prime}(t) &= \langle 0,\; 2,\; 0\rangle \\ + \vec{r}^{\prime\prime\prime}(t) &= \langle 0,\; 0,\; 0\rangle \\[4pt] + \vec{r}^{\prime}\times\vec{r}^{\prime\prime} &= + \begin{vmatrix} + \hat{i} & \hat{j} & \hat{k} \\ + 2 & 2t & 0 \\ + 0 & 2 & 0 + \end{vmatrix} + = \langle 0,0,4\rangle \\[4pt] + (\vec{r}^{\prime}\times\vec{r}^{\prime\prime})\cdot\vec{r}^{\prime\prime\prime} &= \langle 0,0,4\rangle\cdot\langle 0,0,0\rangle = 0 \\[4pt] + \tau &= \frac{(\vec{r}^{\prime}\times\vec{r}^{\prime\prime})\cdot\vec{r}^{\prime\prime\prime}} + {\|\vec{r}^{\prime}\times\vec{r}^{\prime\prime}\|^2} + = \frac{0}{\| \langle0,0,4\rangle\|^2} = 0. + \end{align*} + This corresponds with answer choice A. +} +\qs{}{ + \begin{align*} + \vec{r}(t) &= \langle 3\sin t,\; 3\cos t,\; -t\rangle \\ + \vec{r}^{\prime}(t) &= \langle 3\cos t,\; -3\sin t,\; -1\rangle \\ + \vec{r}^{\prime\prime}(t) &= \langle -3\sin t,\; -3\cos t,\; 0\rangle \\ + \vec{r}^{\prime\prime\prime}(t) &= \langle -3\cos t,\; 3\sin t,\; 0\rangle \\[4pt] + \vec{r}^{\prime}\times\vec{r}^{\prime\prime} &= + \begin{vmatrix} + \hat{i} & \hat{j} & \hat{k} \\ + 3\cos t & -3\sin t & -1 \\ + -3\sin t & -3\cos t & 0 + \end{vmatrix} + = \langle -3\cos t,\; 3\sin t,\; -9\rangle \\[4pt] + \| \vec{r}^{\prime}\times\vec{r}^{\prime\prime}\|^2 &= (-3\cos t)^2+(3\sin t)^2+(-9)^2 = 9+81=90 \\[4pt] + (\vec{r}^{\prime}\times\vec{r}^{\prime\prime})\cdot\vec{r}^{\prime\prime\prime} + &= (-3\cos t)(-3\cos t) + (3\sin t)(3\sin t) + (-9)(0) = 9(\cos^2 t+\sin^2 t)=9 \\[4pt] + \tau &= \frac{9}{90} = \frac{1}{10}. + \end{align*} + This corresponds with answer choice B. +} \end{document}