Spherical coordinates, vector field and dot product

[Rombus]

Homework Statement

Show that the vector fields A = a_r(sin2θ)/r²+2a_θ(sinθ)/r² and B = rcosθa_r+ra_θ are everywhere parallel to each other.

Homework Equations

$\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos(0)$

The Attempt at a Solution

So, if the dot product equals 1. They should be parallel correct?

A={sin(2θ)/(r²),2(sin(θ)/r²),0}
B={rcos(θ),r,0}

if this is the dot product how do I determine the angle between the vectors?
(2 Sin(θ))/r + (Cos(θ) Sin(2 θ))/r

Do i need to transform to rectangular coordinates?

 

[gabbagabbahey]

|A||B|=ABcosθ

You'll want to double check this equation

So, if the dot product equals 1. They should be parallel correct?

Careful, $\mathbf{i}+2\mathbf{j}$ and $\mathbf{i}$ are not parallel, but their dot product is 1. Likewise, $\mathbf{i}+\mathbf{j}$ and $2\mathbf{i}+2\mathbf{j}$ are parallel but their dot product is not equal to 1.

If 2 vector fields are parallel, what can you say about the angle between them at every point? What does the dot product formula then tell you?

 

[Rombus]

Hello, thanks for the reply. I blame lack of sleep on my dot product equation mishap. :zzz:

So, the angle between the vector fields is 90 degrees and the dot product would be 0 correct?

 

[gabbagabbahey]

Hello, thanks for the reply. I blame lack of sleep on my dot product equation mishap.

So, the angle between the vector fields is 90 degrees and the dot product would be 0 correct?

:zzz: a 15 minute nap can sometimes do a world of good for one's studies

If the angle between two vector fields is 90 degrees, then they are perpendicular, not parallel

 

[Rombus]

of course! Okay, so the angle is zero or 180. So upon finding the dot product how would I determine the angle between these two fields from the result of this dot product? (2 Sin(θ))/r + (Cos(θ) Sin(2 θ))/r

Do I just plug in zero for theta?

 

[gabbagabbahey]

of course! Okay, so the angle is zero or 180.

Wouldn't 180 degrees mean the vector fields were anti-parallel?

So upon finding the dot product how would I determine the angle between these two fields from the result of this dot product? (2 Sin(θ))/r + (Cos(θ) Sin(2 θ))/r

Do I just plug in zero for theta?

No, the θ in the equations for your 2 vector fields is either the polar angle (the angle between the position vector and the polar axis) in spherical coordinates, or the azimuthal angle (the angle between the projection of the position vector onto the xy-plane, and the x-axis), depending on which naming convention you are using for spherical coordinates.

That θ is, in general, not the same as the angle between the two vector fields.

If the angle between the two vector fields is 0, then the (correct) dot product equation tells you $\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}||\mathbf{B}|\cos(0)$