Answer
$v(t)=(\sec t\tan t)i+(\sec^2 t)j+\frac{4}{3}k $
$a(t)=(\sec t \tan ^{2}t+\sec^{3}t)i+(2\sec^{2}t \tan t)j$
Speed: $\vert v(\frac{\pi}{6})\vert=2 $
Direction:$=\frac{1}{3}i+\frac{2}{3}j+\frac{2}{3}k$
Velocity: $2(\frac{1}{3}i+\frac{2}{3}j+\frac{2}{3}k)$
Work Step by Step
Calculate $v(t)$ and $a(t)$:
$v(t)=(\sec t\tan t)i+(\sec^2 t)j+\frac{4}{3}k $
$a(t)=(\sec t \tan ^{2}t+\sec^{3}t)i+(2\sec^{2}t \tan t)j$
Speed:
$\left| v\left(\frac{\pi}{6}\right)\right|=\sqrt {\left(\sec \frac{\pi}{6}\tan \frac{\pi}{6}\right)^{2}+\left(\sec^{2}\frac{\pi}{6}\right)^{2}+\left(\frac{4}{3}\right)^{2}}=2 $
Direction:
$\frac{v(\frac{\pi}{6})}{\vert v(\frac{\pi}{6})\vert}=\frac{(\sec \frac{\pi}{6}\tan \frac{\pi}{6})i +(\sec^{2}\frac{\pi}{6})j+\frac{4}{3}k }{2}=\frac{1}{3}i+\frac{2}{3}j+\frac{2}{3}k$
The velocity is:
$v(\frac{\pi}{6})=2(\frac{1}{3}i+\frac{2}{3}j+\frac{2}{3}k)$
