Question

Can someone help me understand this?

Answer 1

I'm reading a book on classical mechanics and I can't quite catch on to what the author is saying.

The scalar product \[\vec r \cdot (d\vec r)= r(d\vec r)_r\]where \[ (d\vec r)_r\] is the projection
of \[d\vec r\] on the vector \[\vec r\] This projection is equal to \[d\vec r\] the
increment of the magnitude of the vector \[\vec r\] Therefore \[\vec r \cdot (d\vec r)= r*dr\]

Answer 2

from the sketch you can see\[\vec r \cdot d\vec r= \left| \vec r \right| \left| d\vec r \right| \cos \theta\]and\[ \left| d\vec r \right| \cos \theta=dr\]hence:\[\vec r \cdot d\vec r= r\:dr\]