I think r(t)=<j,j,k> should be r(t)=<x,y,z>. Take a point r(x,y,z) in 3-space and convert it into a position vector:
r=xi+yj+zk where i, j and k are unit vectors (a unit vector has a length of 1 in a particular direction) in the 3 orthogonal directions. This vector can be written:
r=<x,y,z> is the shorthand way of representing r. The magnitude of r is written |r|=√(x2+y2+z2) and is called distance or length. It's the length of the line joining the point r to the origin (0,0,0).
r(t) makes the position vector dependent on t (a scalar parameter, often representing time).
This implies that each of the components of the vector depends on t, so x, y and z are functions of t.
So we can write r(t)=<x(t),y(t),z(t)> to make it clear that all the components of r depend on t.
The derivative of a position vector with respect to time is dr/dt=<dx/dt,dy/dt,dz/dt>. This derivative is the velocity vector v, so it has magnitude and direction. The magnitude is called speed and can be represented by v=√(dx/dt)2+(dy/dt)2+(dz/dt)2) and is the rate of change in time of the length of the line joining r(x,y,z) to the origin (0,0,0). Note that speed (a scalar) is always a positive quantity. For example, a car travelling along a road at a speed of 30mph heading north-east has no vertical velocity component so z=0 and dz/dt=0; but its component east (which we'll the x direction) is 30/√2mph and its component north (y direction) is also 30/√2mph, making vector v=<30/√2,30/√2,0>, so speed v=|v|=√(450+450+0)=√900=30mph. (If the car were heading south-west (directly opposite to north-east), v=<-30/√2,-30/√2,0> because east is the x-direction, so west is negative; and south is negative with respect to a positive north (y). But the speed is still 30mph. This is what distinguishes speed from velocity.)
The acceleration vector, often represented by a, is the derivative of the velocity vector v.
a=<d2x/dt2,d2y/dt2,d2z/dt2> and its magnitude is √(d2x/dt2)2+(d2y/dt2)2+(d2z/dt2)2). If the car in the example wasn't accelerating the second derivatives would all be zero. If it was accelerating d2z/dt2 would still be zero; but if the car accelerates from 30 to 60mph in 5 seconds then its acceleration is 30/5=6mph/sec, so the components of its acceleration vector would be d2x/dt2=d2y/dt2=6/√2mph/sec because it's accelerating in the direction of travel, north-east.
I hope I've helped you to understand.
