Centripetal vs Radial acceleration

[Amio]

I am reading introductory physics from Serway. Where they say if $$a_r$$ is radial acceleration and $$a_c$$ is centripetal acceleration then $$a_c = v^2/r$$ and $$a_r = -a_c = - v^2/r$$
But aren't the radial and centripetal acceleration same (correct me if I am wrong)? Why is there a minus sign?
The book explains by saying that the negative sign indicates that the direction of centripetal acceleration is towards the center of the circle representing the radius of curvature..
I don't understand this explanation because as the direction of the radial acceleration is also towards the center, shouldn't it be $$a_{radial} = a_{centripetal} ?$$ Why the minus sign?
Would someone please clarify?

 

[D H]

Do you know what spherical coordinates are? The radial component of a vector is the projection of that vector onto the radial unit vector. That vector points away from the origin. Thus the radial component of acceleration is positive if the acceleration vector is pointing away from the origin.

Centripetal acceleration is acceleration toward the center.

It's just a sign convention, nothing else.

 

[Amio]

Thank you.

 

[D H]

The term centripetal acceleration applies only when an object is following a curved path, and by definition the centripetal acceleration is always positive. On the other hand acceleration can have a positive or negative radial component. Consider two electrons moving toward another, for example, with the origin at the center of mass. The radial component of acceleration will be positive because like charges repel.