How to express vector in sigma notation

[reckon]

A vector A, can be expressed in sigma notation as
\sum A_i where i runs from 1 to 3, i.e. A₁ for x coordinate, A₂ for y coordinate and A₃ for z coordinate.

I wonder how to express vector A in polar form using sigma notation. Could anyone share their knowledge to me?

 

[tiny-tim]

Welcome to PF!

Hi reckon ! Welcome to PF!

(have a sigma: ∑ )

A vector A, can be expressed in sigma notation as
\sum A_i where i runs from 1 to 3, i.e. A₁ for x coordinate, A₂ for y coordinate and A₃ for z coordinate.

No, it's ∑ A_ie_i, where the {e_i} are basis vectors.

I wonder how to express vector A in polar form using sigma notation. Could anyone share their knowledge to me?

You'd need to use the basis vectors e_r e_θ and e_φ.

But these are different at each point, and at the origin they aren't even defined.

 

[reckon]

Hahaha.. thanks tiny-tim for ∑ and you even provided with "for copying-pasting" in your signature.

But I'm still a bit unsure because for e_θ e_φ, we only have angle which doesn't have dimension, isn't it? Can we make them as component of a vector?

A=re_r + θe_θ + φe_φ
where r have length dimension while θ and φ have no dimension?

 

[tiny-tim]

Hi reckon!

But I'm still a bit unsure because for e_θ e_φ, we only have angle which doesn't have dimension, isn't it? Can we make them as component of a vector?

A=re_r + θe_θ + φe_φ
where r have length dimension while θ and φ have no dimension?

ah … you obviously haven't come across these before …

no, e_r e_θ and e_φ are all unit vectors (with magnitude one), in the directions of increasing r, increasing θ, and increasing φ, respectively …

so they all have length dimension.

(it is traditional to use the symbol e for unit vectors)

A vector starting at the position (r,θ,φ) will be written as ae_r + be_θ + ce_φ, and a b and c have nothing to do with (r,θ,φ), they are simply the components of the vector in each of the three (perpendicular) directions.

(this would be easier to follow with a diagram, but unfortunately, I can't find a good one, in wikipedia or anywhere else )