TITLE: vector/tensor covariance and contravariance notation
QUESTION [7 upvotes]: As I looked over the Wikipedia article on covariance and contravariance of vectors and $\mathbf{v}=v^i\mathbf{e}_i$ is said as a contravariant vector while $\mathbf{v}=v_i\mathbf{e}^i$ is said as covariant vector (or covector).
However, in the latter part, the article says:

Then the contravariant coordinates of any vector $\mathbf{v}$ can be obtained by the dot product of $\mathbf{v}$ with the contravariant basis vectors: $q^1=\mathbf{v}\cdot \mathbf{e}^1$, $q^2=\mathbf{v}\cdot \mathbf{e}^2$, and $q^3=\mathbf{v}\cdot \mathbf{e}^3$. Likewise, the covariant components of $\mathbf{v}$ can be obtained from the dot product of $\mathbf{v}$ with covariant basis vectors, viz. $q_1=\mathbf{v}\cdot \mathbf{e}_1$, $q_2=\mathbf{v}\cdot \mathbf{e}_2$, and $q_3=\mathbf{v}\cdot \mathbf{e}_3$.

I am getting confused. It seems that the location of the indices (up or down) of contravariant or covariant vectors is different in these two different parts.
Can anyone show me what the heck is this?
Thanks.

REPLY [2 votes]: I'm a bit late to this thread, but I want to clarify one point.  The vector, ${\bf v} = v^i{\bf e}_i = v_i{\bf e}^i$, is neither contravariant nor covariant.  It is the components that are either contravariant ($v^i$) or covariant ($v_i$).
A given vector, ${\bf v}$, can be represented by the components it has in some basis or dual basis, and those components will generally change if the basis is changed, but the vector does not change.  This also extends to tensors with more indices.  So "contravariant" and "covariant" refer to how the components of the object will change when changing from one basis to another, but the object itself does not change.  
In practice, it is common to think about and talk about the components as if they are the vector, which is fine if you just want to calculate something, but the distinction can be important when talking about the abstract properties of the vector.
EDIT: The equality $v^i{\bf e}_i = v_i{\bf e}^i$ is more a correspondance than and equality, as is discussed on this post.