Motion in two dimensions, in a plane can be studied by expressing position, velocity and acceleration as vectors in cartesian coordinates `A=A_(x)hat(i) + A_(y)hat(j)`, where `hat(i)` and `hat(j)` are unit vector along x and y-directions, respectively and `A_(x)` and `A_(y)` are corresponding components of A. Motion can also be studied by expressing vectors in circular polar coordinates as `A= A_(r)hat(r) + A_(theta) hat(theta)`, where `hat(r)=(r)/(r)=cos theta hat(i)+sin theta hat (j)` and `hat(theta)=-sin theta hat(i) + cos theta hat (j)` are unit vectors along direction in which r and `theta` are increasing.
(a) Express `hat(i)` and `hat (j) ` in terms of `hat(r) ` and `hat (theta)`.
(b) Show that both `hat(r)` and `hat(theta)` are unit vectors and are perpendicular to each other.
(c) Show that `(d)/(dt)(hat(r))= omega hat(theta)`, where `omega=(d theta)/(dt)` and `(d)/(dt) (hat(theta))=-thetahat(r)`.
(d) For a particle moving along a spiral given by `r=a theta hat(r)`, where a = 1 (unit), find dimensions of a .
(e) Find velocity and acceleration in polar vector representation for particle moving along spiral described in (d) above.