Question

if i,j,k are unit vectors in cartesian system & e(r), e(θ), e(z) are unit vectors in cylindrical system, i have to show that-

'i'= 'e(r)' cosθ – 'e(θ)' sinθ

'j' = 'e(r)' sinθ+ 'e(r)' cosθ

'k'= 'e(r)'


the quantities in '_ ' are vectors.

I have to prove it by simple geometry & calculus, without using jacobian or linear algebra (basis).