Vector addition in cylindrical coordinates

[yungman]

My question is about vector addition in cylindrical coordinates:
Let A = 2x + y, B = x + 2y. In rectangular coordinates, AB = B-A = -x+y

In cylindrical coordinates, x=rcosθ + θsinθ, y=rsinθ + θcosθ
A =Axx + Ayy, B =Bxx + Byy

Ar = Ax(x.r) + Bx(y.r)=2.236, Aθ = 0. So A = 2.236r
Br = 2.236, Bθ = 0. So B = 2.236r

How do you do vector addition in cylindrical coordinates? A + B = 2.236r +2.236r !

Attached is the hand written file for clearer description.

I don't know how to add the two vectors totally in cylindrical coordinates because the angle information is not apparant. Please tell me what am I doing wrong.
Thanks

 

[maze]

You can always convert to cartesian coords, add, then convert back again

 

[Defennder]

For one thing, you can't just add $$A_r + B_r = 2\sqrt{5}$$. Drawing the diagram for the parallelogram law of vector addition in rectangular basis vectors already shows that the length of the resulting vector is less than the sum of the length of the 2 vectors which are added together (unless we're considering identical vectors). You can picture how this may be done geometrically in the parallelogram addition picture, but it appears to be tedious and unnecessary.

Firstly find the angle between A and the x-axis, as well as B and the x axis. One of this vectors, so to speak, would be leaning "closer" to the x-axis. You can then draw the "triangle law picture" of vector addition and find the length of the resulting vector by using cosine rule. That gives you the value of $$C_r$$ where C is the resulting vector. Then you can use sine rule to find the angle between the resulting vector and the x-axis to give you $$C_{\theta}$$.

But all this is really unnecessary is it not?

 

[yungman]

For one thing, you can't just add $$A_r + B_r = 2\sqrt{5}$$. Drawing the diagram for the parallelogram law of vector addition in rectangular basis vectors already shows that the length of the resulting vector is less than the sum of the length of the 2 vectors which are added together (unless we're considering identical vectors). You can picture how this may be done geometrically in the parallelogram addition picture, but it appears to be tedious and unnecessary.

Firstly find the angle between A and the x-axis, as well as B and the x axis. One of this vectors, so to speak, would be leaning "closer" to the x-axis. You can then draw the "triangle law picture" of vector addition and find the length of the resulting vector by using cosine rule. That gives you the value of $$C_r$$ where C is the resulting vector. Then you can use sine rule to find the angle between the resulting vector and the x-axis to give you $$C_{\theta}$$.

But all this is really unnecessary is it not?

Thanks for the reply. I spent some time on this and conclude vector in curvilinear coordinates cannot be added unless they both belong to the same curvilinear coordinates( origin at the same point.) In my example, the two position vector will translate into two separate curvilinear coordinates by ordinary transformation which cannot be added in the simple manner.

I am learning about math and I am no where good at it. I don't dare to make any conclusion until I run it by you guys, you people are the expert.

If anyone have any comments on what I said, please let me know, I am a self study and I really not sure I understand things correctly.

Thanks