Vectors are often expressed in terms of their components in rectangular coordinates. One common convention is to write these components as an ordered triple, namely $$\vv=(v_x,v_y,v_z)$$ or as a list of components, namely $$\vv=\langle v_x,v_y,v_z \rangle$$ Another popular convention is to call the basis vectors in the $x$, $y$, and $z$ directions $\{\ii,\jj,\kk\}$, respectively, leading to $$\vv=v_x\ii +v_y\jj+ v_z\kk$$ This convention is a historical hangover from attempts to describe electrodynamics with quaternions in the late 1800's! A more recent notation, and the one we will adopt, is to name the basis vectors after the coordinates being used, replacing $\{\ii,\jj,\kk\}$ by $\{\xhat,\yhat,\zhat\}$, and writing $$\vv=v_x\xhat +v_y\yhat+ v_z\zhat$$ That is, we expand vectors in terms of the basis $\{\xhat,\yhat,\zhat\}$. 1)

When using curvilinear coordinates, it is often useful to expand the vector in terms of unit vectors associated with the curvilinear coordinates instead. For example, $\rhat$ is defined to be the unit vector pointing in the direction of increasing $r$ (with the other coordinates held fixed). 2) Unlike the rectangular basis vectors, these vector fields vary from point to point. The adapted basis vectors for cylindrical and spherical coordinates are shown in § {Wrap-Up: Curvilinear Basis Vectors}. Before looking at the figures in that section, you should try to visualize these basis vectors for yourself. Imagine that the origin of coordinates is behind you in the corner of the room you are sitting in, on the floor, on the left-hand side. Think of your right shoulder as a point in space. You should be able to point successively in each of the directions $\xhat$, $\yhat$, and $\zhat$ in rectangular coordinates, $\rhat$, $\phat$, and $\zhat$ in cylindrical coordinates and $\rhat$, $\that$, and $\phat$ in spherical coordinates.