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tylerscott

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## Homework Statement

In the oblique coordinate system K' defined in class the position vector r′ can be written as:

[itex]r'=a\hat{e'}_{1}+b\hat{e'}_{2}[/itex]

Are a and b the covariant (perpendicular) or contravariant (parallel) components of r′? Why? Give an explanation based on vectors’ properties and another based on tensors’ properties.

## Homework Equations

[itex]\hat{e'}_1=\hat{e}_1[/itex]

[itex]\hat{e'}_2=e_1cos(\alpha )+e_2sin(\alpha )[/itex]

[itex]{x'}_{1\perp}={x}_1[/itex]

[itex]{x'}_{2\perp}=x_1cos(\alpha )+x_2sin(\alpha )[/itex]

## The Attempt at a Solution

My best effort is that a and b are the covariant components of r' since they are the perpendicular projects of r' onto the [itex]\hat{e'}_1[/itex] and [itex]\hat{e'}_2[/itex] basis vectors, so they're essentially equivalent to [itex]{x'}_{1\perp}[/itex] and [itex]{x'}_{2\perp}[/itex]. So I think this would be my vector solution for the problem, but I don't know exactly how to represent it. As for the an explanation based on the tensor's properties, I don't even know where to start...

Thanks in advance.

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