We have three orthonormal vectors [tex] \vec i_1 , \vec i_2, \vec i_3 [/tex] , and we know wich are the components of an arbitrary vector [tex] \vec A[/tex] in this base, explicitly:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \vec A = (\vec A \bullet \vec i_1) \vec i_1 + (\vec A \bullet \vec i_2) \vec i_2 + (\vec A \bullet \vec i_3) \vec i_3 [/tex]

If now we want to generalize to a base that is orthogonal , but is NOT normalized, we can divide by the modulus, and we came back to the first case, so we have:

[tex] \vec i_1 = \frac{\vec e_1}{e_1} , \vec i_2 = \frac{\vec e_2}{e_2}, \vec i_3 = \frac{\vec e_3}{e_3} [/tex]

So now the expression of [tex] \vec A[/tex] is:

[tex] \vec A = \frac{\vec A \bullet \vec e_1}{e_1^2} \vec e_1 + \frac{\vec A \bullet \vec e_2}{e_2^2} \vec e_2 + \frac{\vec A \bullet \vec e_3}{e_3^2} \vec e_3 [/tex] Equation 1

In the next case the base will be noncoplanar, not orthogonal and the vectors won't be normalized, I am following the book from Borisenko and Taraponov about tensor calculus, they introduce the reciprocal bases to solve this problem to arrive at this expression:

[tex] \vec A = (\vec A \bullet \vec e^1) \vec e_1 + (\vec A \bullet \vec e^2) \vec e_2 + (\vec A \bullet \vec e^3) \vec e_3 [/tex]

Where the reciprocal base is defined as:

[tex] \vec e^1 = \frac{\vec e_2 \times \vec e_3}{\vec e_1 \bullet (\vec e_2 \times \vec e_3 )}[/tex]

e^2, and e^3 have a similar definition in terms of e_1, e_2 and e_3.

Well the question is ¿why do we need to introduce the reciprocal base to solve this problem? ¿what is the benefit of using it?.

I don't know why you don't calculate the components of A using equation 1 . You can project A to the vector e_1, e_2 and e_3 using the dot product and you find the components of A ,in a similar way as you have done before with the other basis ¿is there something that impede you to make the calculation in this way?.

And you have another option, simply using this equality:

[tex] \vec A = \vec A [/tex]

First A expressed in the orthonormal base , and then expressed in the more general noncoplanar, nonorthogonal and not normalized :

[tex] (\vec A \bullet \vec i_1) \vec i_1 + (\vec A \bullet \vec i_2) \vec i_2 + (\vec A \bullet \vec i_3) \vec i_3 = A^1 \vec e_1 + A^2 \vec e_2 + A^3 \vec e_3 [/tex]

And you can find three equations to solve, so you wold not need introduce other base ¿is anything wrong in this last equation?

The book then introduce the concept of covariant and contravariant components of a vector, and I would like to understand why do you need the reciprocal base, and what it is.

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# Vector expressed in a basis noncoplanar, neither orthogonal nor of unit length

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