- #1

George Keeling

Gold Member

- 156

- 39

## Homework Statement

I am studying co- and contra- variant vectors and I found the video at youtube.com/watch?v=8vBfTyBPu-4 very useful. It discusses the slanted coordinate system above where the X, Y axes are at an angle of α. One can get the components of

**v**either by dropping perpendiculars to the axes (v

_{i}) or by dropping a line parallel to the other axis (v

^{i}). These give correct results for the norm v

^{i}v

_{i}and the dot product v

^{i}w

_{i}. (I have not shown

**w**). So the v

^{i}are called contravariant and the v

_{i}are called covariant. According to the video Dirac thought this was a great example.

But both v

_{i}and v

^{i}contra-vary with a change of scale of the basis vectors. This contradicts some definitions of contravariant and covariant components, e.g. this one on Wikipedia. These definitions say that covariant components co-vary with a change of scale.

Is there a simple resolution to this apparent contradiction?

## Homework Equations

Norm and dot product were calculated by expressing v

_{i}and v

^{i}in Cartesian coordinates. Quite a lot of equations! Along the way we found the metric for the slanted coordinate system.

## The Attempt at a Solution

We can demonstrate the problem by drawing basis vectors on the diagram and then another set double their size.