Thread: 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 k-direction), 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 end-points) and the k-direction.


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
(the square of the distance from the particle to the origin)


in other words find t such that and

NB

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

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