Can you refresh my memory how you add a negative vector? Is this simply the same as just subtracting the two vectors? Is subtraction what we are after when trying to calculate the missing phase.

Adding a negative vector is the same as subtracting a positive vector... the signs are the same as basic arithmetic. So if we retain "b" as the common connection of PT's and base are voltage vectors as such, yes we would subtract vectors to calculate the non-sampled voltage.

To change from for instance a Vcb to Vbc all we simply do is reverse the direction of the vector correct? So what would happen instead of adding -Vcb as you mentioned we added Vbc? Would we get the same answer. Can you explain quickly the reason why the particular vectors are used as you mentioned for calculating.

Yes, reversing the letters (reference point) is the same as negating or reversing the direction of the vector. If instead of adding a positive vector we added its negated vector, we would not get the same answer. Both the magnitude and direction would be wrong for any two vectors that are not at 90? to each other... and when they are 90? to each other, the magnitude would be correct but the direction would be wrong.

Using these two vectors over any of the other two possible combinations is just an arbitrary choice when using open-delta PT's.

So I guess for calculating the L-N voltages it just take half of the measured angle between the vectors and calcultates magnigutes to intersection point?

I'm fairly certain that's how it's done when L-N voltages are not sampled.

Why must the third L-L voltage be calculated with this PT arrangement? If it can physically be measured with a meter ( I can measure 120V between all 3 L-L connections) then why must it be calcuated as opposed to being measured directly?

Well in a sense, all voltage measurements are calculated... just with this arrangement we are using three points rather than two.

For example, say we connected two 12VDC batteries in series. We can measure across each and get 12 volts and we know from the dual battery configuration that to measure across both we will get 24 volts. With two meters we can measure each battery simultaneously, and with a third meter, we could verify that the other two measures add up to the third's.

The only difference here is that these are AC voltages, at the same frequency but out-of-phase to each other. By determining through calculation (actually by sample comparison) we can determine the extent of the out-of-phase relationship, and then through calculation the measure of third voltage and its phase relationship to the other two. Experience has taught us this is a viable method of measuring the third unsampled voltage.