Math Challenges

Submissions for Problem #25

Given the vector function r(t), find the unit tangent vector T(t) at the point with the given value of t

\[ \displaylines{\mathbf{r\left(t\right)}=t\mathbf{i}+2\sin\left(t\right)\mathbf{j}+3\cos\left(t\right)\mathbf{k}\\ t=\frac{\pi}{6}} \]

zapwai

Solution: \( \displaylines{\mathbf{r}^{\prime}\left(t\right)=\lt1,2\cos t,-3\sin t\gt\Rightarrow\mathbf{T\left(t\right)}=\frac{\mathbf{r}^{\prime}\left(t\right)}{\left|\mathbf{r}^{\prime}\left(t\right)\right|}=\frac{\lt1,2\cos t,-3\sin t\gt}{\sqrt{1+4\cos^2t+9\sin^2t}}\\ \mathbf{T\left(\frac{\pi}{6}\right)}=\frac{4\lt1,\sqrt3,-\frac32\gt}{25}} \)