Below is a picture of a basic example from my book. It\'s probably horribly simp

Question

Below is a picture of a basic example from my book.  It's probably horribly simple and im just having a brain lapse here but howcome PLF = 1/2.   I'm guessing im forgetting some rule about vector multiplication or something but if anyone could fill me in on why it is 1/2 I'd be greatful.  Please make sure to give a detailed explanation in your response.  I will rate quickly :)

Find the polarization loss factor (PLF). Solution: For the incident wave and for the antenna The PLF is then equal to which in dB is equal to

Explanation / Answer

Its simple.

Rho-a and Rho-w are two vectors.

PLF is Square of dot product between the two vectors Rho-a and Rho-b

Vector Rho-a is having only ax component, where as Vector Rho-b having both ax and ay components.

ax. ax = 1, ax. ay = 0

so, the dot product gives only 1/sqrt(2) . Further square of 1/sqrt(2) gives 1/2.

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