Vectors are often expressed in terms of their components in rectangular coordinates. That is, we expand vectors in terms of the basis $\{\xhat,\yhat,\zhat\}$, which are the unit vectors pointing in the $x$, $y$, and $z$ directions, respectively. 1) $$\vv=v_x\xhat +v_y\yhat+ v_z\zhat$$ The common convention of writing a vector in terms of an ordered triple $$\vv=(v_x,v_y,v_z)$$ is just a short-hand that avoids explicitly writing the basis vectors. Another popular convention is to call the basis vectors $\{\ii,\jj,\kk\}$, respectively. $$\vv=v_x\ii +v_y\jj+ v_z\kk$$ This convention is an historical hang-over from attempts to describe electrodynamics with quaternions in the late 1800's!

When using curvilinear coordinates, it is often useful to instead expand the vector in terms of unit vectors associated with the curvilinear coordinates. 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 $\ii$, $\jj$, and $\kk$ in rectangular coordinates, $\rhat$, $\phat$, and $\zhat$ in cylindrical coordinates and $\rhat$, $\that$, and $\phat$ in spherical coordinates.