High School How to find the dual basis vector for the following

[LSMOG]

e_i=i+j+2vk , how to find the dual basis vector if the above is a natural base?

 

[Orodruin]

It is not sufficient to know one of the basis vectors to deduce the corresponding dual basis. You must know the entire basis to find its dual.

 

[LSMOG]

It is not sufficient to know one of the basis vectors to deduce the corresponding dual basis. You must know the entire basis to find its dual.

Thak you. They are
e₁=i+j+2vk
e₂=i-j+2uk
e₃=k
How to find their dual basis?

 

[Orodruin]

What should the dual basis satisfy? How can you find vectors that satisfy these conditions?

 

[LSMOG]

What should the dual basis satisfy? How can you find vectors that satisfy these conditions?

Mmm, my book does not explain. It just says, the dual basis are...I can't figure it out

 

[LSMOG]

What should the dual basis satisfy? How can you find vectors that satisfy these conditions?

I got. I use the ∇ operator to find the gradient of each natural base

 

[Orodruin]

Mmm, my book does not explain. It just says, the dual basis are...I can't figure it out

This is not helping much. Please do not refer to "my book" without stating which book you are using.

I got. I use the ∇ operator to find the gradient of each natural base

The gradient of a coordinate function gives you the corresponding dual basis vector in the same way as the tangent vector basis are the tangent vectors of the coordinate lines. However, if you are just given the tangent vector basis and not the actual coordinate functions, this is not a viable way forward since you do not have the coordinate functions to take the gradient of. (Do you have the coordinate functions? Please reproduce the entire problem exactly as stated.) However, there are certain relations between the dual basis vectors and the tangent vector basis that need to be satisfied and that you can use to deduce the dual basis from the tangent vector basis.

 

[alejandromeira]

Do you know the conditions of a dual basis?
Have you built the metric and the inverse of the metic?
what is the dot product between the vectors of the covariant basis and the vectors of the contravariant basis?

I believe that these questions are relevants for your project.
Ok, I'm not an expert, then if I say something wrong I hope someone corrects me.

 

[LSMOG]

Do you know the conditions of a dual basis?
Have you built the metric and the inverse of the metic?
what is the dot product between the vectors of the covariant basis and the vectors of the contravariant basis?

I believe that these questions are relevants for your project.
Ok, I'm not an expert, then if I say something wrong I hope someone corrects me.

I think the product is Kronecker delta

 

[Orodruin]

I think the product is Kronecker delta

So how many relations does that give you? How many unknowns do you have?

Edit: And more importantly, what does it tell you about, for example, the relation between $\vec e^1$ and $\vec e_2$ and $\vec e_3$?

 

[alejandromeira]

I think the product is Kronecker delta

Yes.
Dot product, and then Kronecker delta has relevant consecuences in the orientation and the length of the vectors of the dual basis.
You can compute the dual basis without the metric like I think Orodruin suggest you, based in geometry. But raise and lower indices, is also a routinary task with the metric.