The basis vectors adapted to a particular coordinate system are perpendicular to each other at every point. In particular, \begin{eqnarray} \rhat\cdot\phat &= \phat\cdot\zhat &= \zhat\cdot\rhat = 0 \qquad\hbox{(cylindrical)} \\ \rhat\cdot\that &= \that\cdot\phat &= \phat\cdot\rhat = 0 \qquad\hbox{(spherical)} \\ \end{eqnarray} Figure 2 of § {Curvilinear Basis Vectors} shows this orthogonality in the case of polar basis vectors. These basis vectors are also normalized; they are unit vectors: \begin{eqnarray} \rhat\cdot\rhat &= \phat\cdot\phat &= \zhat\cdot\zhat = 1 \qquad\hbox{(cylindrical)} \\ \rhat\cdot\rhat &= \that\cdot\that &= \phat\cdot\phat = 1 \qquad\hbox{(spherical)} \\ \end{eqnarray} A basis with both of these properties is called orthonormal.

We only work with orthogonal coordinates, such as rectangular, cylindrical, or spherical coordinates. Each such coordinate system is orthogonal because the three coordinates point in mutually orthogonal directions. Choosing our adapted basis in each case to consist of unit vectors automatically yields an orthonormal basis.

Arbitrary vectors can be expanded in terms of a basis; this is why they are called basis vectors to begin with. It is particularly easy to expand an arbitrary vector field $\vv$ in terms of an orthonormal basis. The vectors in an orthonormal basis are mutually orthogonal, so it is just a matter of figuring out how much of $\vv$ points in each of those directions.

For example, we have \begin{eqnarray} \vv &=& v_x \,\xhat + v_y \,\yhat + v_z \,\zhat \qquad\hbox{(rectangular)} \\ &=& v_r \,\rhat + v_\phi \,\phat + v_z \,\zhat \qquad\hbox{(cylindrical)} \\ &=& v_r \,\rhat + v_\theta \,\that + v_\phi \,\phat \qquad\hbox{(spherical)} \\ \end{eqnarray}

Since the basis is orthonormal, it is easy to compute the components geometrically. For instance, \begin{equation} v_\phi = \vv\cdot\phat \end{equation} with similar expressions holding for the other components.

This strategy can be very effective when you are working with one or more vectors whose tails all lie at the same point. However, you must be very careful anytime you are adding or subtracting (or integrating, which is just a continuous form of adding) vectors expressed in curvilinear basis vectors. For example, consider the vector field \begin{equation} \vv = r \,\phat \end{equation} in polar coordinates, and let $\vv(P_1)$ denote the vector field $\vv$ evaluated at the point $P_1$. Then \begin{eqnarray} \vv(P_1)+\vv(P_2) &=& r_1\,\phat_{P_1}+ r_2\,\phat_{P_2}\\\nonumber &\ne& (r_1+r_2)\, \phat\qquad\qquad\hbox{(incorrect)} \end{eqnarray} The last line is not correct because $\phat_{P_1}\ne\phat_{P_2}$, so the coefficients of the basis vectors cannot be factored out. The only way out of this dilemma is to rewrite the curvilinear basis vectors in terms of rectangular basis vectors. Since rectangular basis vectors do not change from point to point, they can be factored out of the relevant expressions.