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I had this small confusion -
The components of a vector field representing a physical quantity must have the same physical dimension right? for example- the radius vector has the unit of length along all three components and so it does represent the position of an object.
What about
$$c(r(r) + cos(\phi) (\phi) + z(k))$$ ?

This can't represent any physical quantity right? whatever the dimension of constant 'c' may be.

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  • $\begingroup$ Related: physics.stackexchange.com/q/12333/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Feb 6, 2015 at 11:59
  • $\begingroup$ Are the $(r)$, $(\phi)$ and $(k)$ terms supposed to be unit vectors? $\endgroup$
    – Kyle Kanos
    Commented Feb 6, 2015 at 13:35
  • $\begingroup$ Yes,(r), (ϕ) and (k) are the unit vectors in the cylindrical coordinate system $\endgroup$
    – Praveen Sriram
    Commented Feb 6, 2015 at 13:47

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I think the expression you are trying to write is $$c \left( r \cdot\hat r + \cos \theta \cdot \hat \theta + z \cdot\hat k \right)$$

As you say this will not represent a physical quantity since since $\cos \theta$ is dimensionless and $r$, $k$ have dimensions of length.

The expression
$$c \left( r \cdot\hat r + z \cdot\hat k \right)$$ would be an allowed physical quantity ( which quantity would depend on the dimensions of $c$ ) and so would also $c \cos \theta \cdot \hat \theta $ be.

Remember that the unit vectors are dimensionless quantities $\hat r = \frac{\vec r}{r}$ so all the dimensionality is carried by their pre-factors.

If that dimensionality is the same for all terms, then the quantity makes mathematical (and physical) sense.

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