How to Compute Dot Products in Integrals?

[nos]

Hi everyone,

I'm trying to understand the integral on http://www.phys.lsu.edu/~jarrell/COURSES/ELECTRODYNAMICS/Chap14/chap14.pdf (page 14)

I get all the steps except the how to get from eq. 48 to eq 49. I'm not really sure how to compute all the dot products. He let's the angle between n and β be θ
and angle between β and $\dot{β}$ be θ(0)

Attempt at solution:

$n\cdotβ=βcos(θ)$
$β\cdot\dot{β}=β\dot{β}cos(θ(0))$
$n\cdot\dot{β}=\dot{β}cos(θ-θ(0))$?

If this is correct, do I proceed by applying the difference identity of cosine in the last dot product and then square the whole thing? There are going to be a lot of terms, so before wasting more time on expanding, let's first see if what I'm doing is in fact the right way to do this integral.

Many thanks!

 

[haruspex]

$n\cdotβ=βcos(θ)$
$β\cdot\dot{β}=β\dot{β}cos(θ(0))$
$n\cdot\dot{β}=\dot{β}cos(θ-θ(0))$?

You could only do something like that if you know the three vectors are coplanar. Are they?

 

[nos]

Yes that is what's been troubling me. I am not sure. But how else do you go from eq 48 to eq 49?

 

[haruspex]

Sorry, I can't follow it either. I strongly suspect a typo, like a β that should be a β-dot or vv., but I haven't been able to find a simple explanation.

 

[nos]

Oh well, thanks very much anyway for replying:)