Complicated path question - 2D vectors

[Kawakaze]

Homework Statement

A model car of mass m moves on a horizontal track so that its
displacement r from a fixed origin at time t is given by

r(t)=2cos(t)i + sin(2t)/sqrt2j where 0<t<2pi

(a) If r is the position vector of a particle, explain the physical
significance of r . r and show how d(r.r)/dt can be used to determine
the maximum distance of the car from the origin.

(b) Show that the car passes through the origin exactly twice in the time interval and determine the two times.

(c) Determine the velocity and acceleration of the car at time t.

(d) Determine the times at which the velocity of the car is perpendicular
to its displacement and the position vector of the car at these times.

The Attempt at a Solution

(a) Unsure of significance, but its pretty clear that when d(r.r)/dt = 0 you have your maximum and minimum distances from the origin.

(b) No idea how to prove this but I guess its obvious that you would have two solutions in the interval 0<t<2pi.

(c) No problem, differentiate once for velocity and twice for acceleration.

(d) This looks like a fun part, but no idea how to calculate this. You want the vectors to be at 90 degrees to each other.

Help please! :)

 

[Muphrid]

Remember that two vectors are perpendicular when their dot product is zero.

 

[Kawakaze]

I know this is what i need to do, is there any worked examples online you know of?

Also still no idea about part (a)