Defining Vectors in Cylindrical Coordinates: Permissible Origin Point?

[Phymath]

is it leagal to define a vector with respect to the orgin in cylindrical coords? can a position vector to a point such as...(a, pi/4, pi/3) can u define a position vector $$<a, pi/4, pi/3>_o, o = (0,0,0)?$$

 

[MathStudent]

A vector in cylindrical coordinates is just an extension of polar coordinates ($r, \theta$) in the xy plane where we ad a third component $z$ that corresponds to the regular third rectangular coordinate. Thus a point ($r, \theta, z$), represents a position vector with end point ($r, \theta, z$).

 

[Phymath]

yes that is true but how would you write the vector, with unit vectors (which we know to be dependent to the point the vectors are written from) ie. the first example i wrote has an end point which seems to be the same as the vector with respect to the point (0,0,0) but with unit vectors how would that be expressed (let e_n be unit vector in the nth dimmension) $$\vec{r_o} = ae_r_o + \pi /4e_\theta _o + \pi /3 e_z_o$$ but isn't e_r_o has no direction cause its refrence is (0,0,0) or does it? is my question

 

[Hurkyl]

You can represent a vector however you like. Just remember you have to represent their addition correctly. (e.g. you cannot just do component-wise addition of their cylindrical coordinates)

 

[Phymath]

right, thank you

 

[MathStudent]

Hurkyl, were you just promoted from mentor to super mentor?

 

[Hurkyl]

Er, no... I'm just mild mannered mathematician Clark Kent...