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 For what value(s) of $t$ is $\left<3tt^3,2t^2\right>$ a vertical vector?
 For what value(s) of $t$ is $\left<1\cos t,\sqrt3+\sin t\right>$ the longest? What is the greatest length, and what vector(s) achieve it?
[Hint: It might make it easier to note that since the length is never negative, it will be longest when its square is longest. Note also that since the length is a number that depends on $t$, it is a singlevariable function, so you can use singlevariable calculus to determine the greatest length. Finally, don't forget that $\tan(t)={\sin(t)\over\cos(t)}$.]

