Shrinking and turning arrows

[BiBi]

In Richard Feynman's book QED, when writing about multiplying arrows of photons (on page 61 if you have the book handy)he shrinks and turns them at specific numbers, and though I understand why and how it's effective, I don't understand how he determined to shrink and turn the arrows by those specific amounts. If anyone knows, when shrinking and turning arrows, how you come up with those amounts and numbers I would be grateful.

 

[selfAdjoint]

The shrinking and turning of the arrows is a feature of the complex arithmetic that Feynman didn't want to get into, which is why he chose to speak of the amplitudes as "little arrows" rather than complex numbers. If you have two complex numbers they can be written $$u cos \theta + iu sin \theta$$ and $$v cos \phi + iv sin \phi$$, where u and v give the lengths and the angles are the ones the two vectors make with the real axis.
If you multiply them you get after simplification $$uv cos(\theta + \phi) + iuv sin (\theta + \phi)$$. So the product has length equal to the products of the two factors and its angle is the sum of theirs. Since the lengths were less than 1 to begin with, the product uv is smaller still, therefore the product is shorter than the factors. And the sum of the angles gives the rotation.

 

[BiBi]

thank you so much! I've been asking my friends for almost a week now, but no one had done QED for a while, if at all. thanks again.