To determine if two vectors are within 90° of each other we simply look for a positive result from the dot product.
$\vec{p} \cdot \vec{q} = \left | \vec{p} \right |\left | \vec{q} \right |\cos\theta $
On pdf page 4 of this paper, they are using a different approach using conjugation that is supposed to be numerically more efficient. If their result, $T$, is positive then the two vectors are within 90° of each other.
$T = Re[\vec{p}*\vec{q^*}]$
Example: $\vec{p}=+j25$, $\vec{q}=-j40$
Dot product approach: $\vec{p} \cdot \vec{q} = (25)(40)\cos180°=-1,000$
Paper approach: $T = Re[(+j25)(+j40)]=-1,000$
So, both approaches produce the exact same result and show the vectors are not within 90° of each other.
My question: What is the derivation for the "paper approach" equation? I can't find anything describing its derivation and would like to understand how it was arrived at.
thanks, russ
Consider the following two vectors: $\vec{x} = (a, b)$ and $\vec{y}=(c,d)$ now imagine for a second that those pairs of numbers represent complex numbers $\hat{x} = a + ib$ and $\hat{y} = c + id$.
Then $\hat{x} \hat{y}^*=(ac + bd) + i (bc - ad)$ taking the real part of that is just $ac + bd$.
On the other hand the dot product $\vec{x}\cdot\vec{y}=(a,b)\cdot(c,d)=ac+bd$ so the two produce the same result.