Hi everyone, I am trying to self study some general relativity however I met some problem in the contravarient and covarient basis. In the lecture, or you can also find it on wiki page 'curvilinear coordinates', the lecturer introduced the tangential vector ei =∂r/∂xi and the gradient vector ei =del xi and showed them to have a relation of ei dot ej = δij . I totally follow and agree the algebraic steps, both on the lecture and on the wiki page. However I cannot figure it out GEOMETRICALLY ! Let consider a 2D curvilinear coordinates, Given we have two tangential basis e1, e2 which are not parallel. How can one geometrically (by drawing and inspection) contruct a third vector that is parallel to e1 but orthogonal to e2 ? However from algebraically we showed the del x1 is the wanted vector !! Lastly, I tried to explicitly see what is going on in the polar coordinate (we are familiar with polar coordinate) however the tangential vector er and eθ are orthogonal... Appreciate you kindly help.