(a) Let r(t) be the position vector in two dimensions dependent on time t. Calculate the time derivative of the unit vector ^r and show directly that this derviative is orthogonal to ^r using explictly the components of these vectors. The vector ^r and the unit vector in the direction of _^r can be chosen as the vectors de¯ning the axes of the polar coordinate system. Draw a single picture showing both the
Cartesian and polar coordinate systems with the unit vectors de¯ning the axes of both systems explicitly marked.
(b) Show using the general rules of di®erentiation that the time derivative of an abitrary (multidimensional) vector of constant length is orthogonal to the vector.