covariant and contravariant vectors

[geordief]

If we have a vector that we  want to define using a pair of  base vectors that do not meet at a 90 degree angle I understand that one way is to  combine these basis vectors  and, adding a coefficient to each  form  a parallelogram which has the  vector as a "destination".

They are contravariant vectors.

 

When it comes to the covariant vectors  it seems that we have to follow these  contravariant vectors until a perpendicular from them would ,if drawn go through the vector we are trying to describe.

 

The question I want to ask is  whether these covariant vectors add in the same way as the contravariant ones.

 

Is is possible to "follow" 2 covariant bases (with appropriate coefficients on each) to ""arrive" at a particular vector  or are they used in a different way **such as  for example  providing an alternative formula for the length of a vector when  their coefficient are "spliced into" the contravariant coefficients for the same vector?

(the x contravariant coefficient  being multiplied by the x covariant coefficient  and their sum by added to the   product of the  y  contravariant and covariant coeficients  to give an alternative to  the squared length of the  vector given by the pythagoran theorem in a cartesian system)

 

**I can't seem to construct a parallelogram using covariant vectors   with the desired   vector at the apex

 

 

Edited February 22, 2019 by geordief