I always thought that a set of "standard basis vectors", meant a set of orthonormal vectors such that a given point in the coordinate system is just a linear combination of those vectors. In Cartesian coordinates, the point (2,1,1) for instance, is just the linear combination 2i + 1j + 1k.

However, my vector calculus book defines the standard basis coordinates for polar coordinates as the set of unit vectors where one vector goes in the r-direction and the other in the theta-direction. I understand that this set will be an orthonormal basis, but is it correct to label them as "standard basis vectors" ? The polar coordinates of the given point, does NOT equal the corresponding linear combination of the alleged "standard basis vectors". In fact, it will just equal a scaled version of the vector in the r-direction.

Any help clarifying this would be greatly appreciated! =))