Contravariant and covariant vectors

[Sonia AS]

I know if the number of coordinates are same in both the old and new frame then A.B=A`.B` . But if the number of coordinates are not same in both old and new frame then A.B=0 means that both the vectors A and B are perpendicular. Why is it so that if the number of coordinates of both the frames are not same, then both the vectors must be perpendicular.

 

[andrewkirk]

The choice of frame has no effect on the dot product. It is always zero for two perpendicular vectors because the definition of perpendicularity is for the dot product to be zero.
The dot product is not defined for vectors with different dimensionality. It is necessary to embed the lower dimensional vector in the higher-dimensional space of the other vector (or to embed both in a third space) in order to obtain a dot product. The embedding may not necessarily be unique, so the dot product may not necessarily be unique.

 

[HallsofIvy]

I know if the number of coordinates are same in both the old and new frame then A.B=A`.B` . But if the number of coordinates are not same in both old and new frame then A.B=0 means that both the vectors A and B are perpendicular. Why is it so that if the number of coordinates of both the frames are not same, then both the vectors must be perpendicular.

What, exactly, do you mean by "number of coordinates"? In any frame, the "number of coordinates" should be equal to the dimension of the space which is independent of the choice of frame. I don't see how it is possible to have "the number of coordinates not the same in both old and new frame".

 

[BiGyElLoWhAt]

I think he means something to the effect of $\vec{A}\cdot \vec{B}$ with $\vec{A} \in \mathbb{R}^2$ and $\vec{B} \in \mathbb{R}^3$

 

[BiGyElLoWhAt]

So replace frames with coordinate systems.

 

[Fredrik]

I think he means something to the effect of $\vec{A}\cdot \vec{B}$ with $\vec{A} \in \mathbb{R}^2$ and $\vec{B} \in \mathbb{R}^3$

Maybe. But then the dot product of $\vec A$ and $\vec B$ is undefined.

 

[BiGyElLoWhAt]

True. But that's the best way that I can interpret what the OP means.

 

[WWGD]

What, exactly, do you mean by "number of coordinates"? In any frame, the "number of coordinates" should be equal to the dimension of the space which is independent of the choice of frame. I don't see how it is possible to have "the number of coordinates not the same in both old and new frame".

Just to add that , if the dimension of the embedded object or subobject is lower than that of the ambient space of dimension n, the coordinates can be parametrized with fewer than n variables, e.g., a curve in space being parametrized by a single variable.