The statement that if the position and velocity have opposite signs, the particle is moving towards the origin is not universally true. It depends on the reference frame and the specific definition of position and velocity being used.
In classical mechanics, position and velocity are vectors that have both magnitude and direction. The position vector points from the origin to the current location of the particle, while the velocity vector describes the rate and direction of change of the position with respect to time.
If we consider a one-dimensional scenario along a straight line, we can define the positive direction as one direction and the negative direction as the opposite direction. In this case, if the particle's position is negative and its velocity is positive, it means that the particle is moving in the positive direction away from the origin. Similarly, if the position is positive and the velocity is negative, the particle is moving in the negative direction towards the origin.
However, it's important to note that in a more general context or in different coordinate systems, the interpretation of positive and negative values may change. The direction of motion cannot be solely determined by the signs of position and velocity without considering the reference frame and coordinate system being used.
So, while there may be cases where a particle with opposite signs of position and velocity moves towards the origin, it's not a general rule that applies in all situations or coordinate systems. The specific dynamics of the system, including forces and accelerations, must be considered to determine the direction of motion accurately.