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 )

 

[CellCoree]

Expressing a vector in polar form using sigma notation can be done by converting the vector's components from Cartesian coordinates (x, y, z) to polar coordinates (r, θ, φ). The sigma notation would then be \sum Ai where i runs from 1 to 3, with A1 representing the magnitude (r), A2 representing the angle in the xy-plane (θ), and A3 representing the angle from the z-axis (φ). This representation can also be written as A = \sum Ar(cosθ, sinθ, cosφ, sinφ). It is important to note that the angle θ is measured counterclockwise from the positive x-axis, and φ is measured from the positive z-axis. I hope this helps in your understanding of expressing a vector in polar form using sigma notation.