
particle question
Not sure if this is in the right forum
A moving particle at time t ∈ [0, 10] (seconds) has position vector in metres from the origin (0, 0, 0) given by the vector function r(t) = (10 − t)i + (t 2 − 10t)j + sin tk.
Describe the path of the particle, as seen from above (the positive kdirection), and also describe it in three dimensions.
Find the curvature of the path, at t = 2π ≈ 6.28 seconds.
Find the angle between the path (at start and endpoints) and the kdirection.
And this one too, i think i have to optimise the square but i don't think i have done it right
A particle’s path, in two dimensions, is described by its position vector (in metres and time t ∈ [1, 2] seconds) relative to point (0, 0, 0) by r(t) = (2t + 1)i + (4 − t^2 )j.
Find the value of t∗ at which the particle has greatest distance from (0, 0, 0).
Show that at position r(t∗ ), the particles velocity is not perpendicular to r(t∗ ).
Thanks heaps if you can help i can't get the right answers

I'm not sure about the first question but for the second you have to maximise
$\displaystyle r(t)^2$ (the square of the distance from the particle to the origin)
in other words find t such that $\displaystyle \frac{d r(t)^2}{dt}=0$ and $\displaystyle \frac{d^2r(t)^2}{dt^2}<0$
NB $\displaystyle r(t)^2=(2t+1)^2+(4t^2)^2$
for the second part find r(t) at the time you found above then find the velocity of the particle v(t) at the time you found above
NB $\displaystyle v(t)=\frac{d}{dt}(r(t))$
Remember if two vectors are perpendicular then there dot product=0 so using those two vectors find it and show it doesn't equal 0