$r(t)=(t-sint)i+(1-cost)jis$ the position $of$ a particle $in$ the $xy-$ plane $att=.$

Find the particle's velocity vector, speed, and acceleration vector $att=.$

$r(t)=ti+tj+\sqrt[\phantom{2}]{9-t^2}kis$ the position $of$ a particle $in$ space $att=0.$ Find the

particle's velocity vector, speed, and acceleration vector $att=0.$ Next, find the

particle's direction $of$ motion $att=0.$ Finally, write the particle's velocity $att=0$

$as$ the product $of$ its speed and direction.

Find the parametric equations for the line that $is$ tangent $tor(t)=(cost+tsint)i$

$+(sint-tcost)j+tkatt=\frac{}{2}.$

Evaluate the indefinite integral: ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(lnti+\frac{1}{t}j+k)dt.$

Evaluate the integral: ${\displaystyle \underset{1}{\overset{ln3}{}}}[t{e}^{t}i+e^{t}j+(lnt)k]dt.$

Solve the given initial value problem for $ras$ a vector function $oft.$

$\frac{d^{2}r}{d{t}^2}=-4costj-3sintk,\frac{dr}{dt}{|}_t=0=3k$ and $r(0)=4j$

please answer all questions.