vectors question a level maths

Try associate "modulus/absolute values" with "distances". Minus inside modulus means "distance between".
So, in English, the equation reads "The distance between (arbitrary point) p and the point with position vector 1/2(a+b) is 1/2|a-b|".

Does that help? It's a "write down" question after all...

thank you, I was thinking along similar lines but I interpreted position vector of p to be 1/2 (a-b) + 1/2(a+b) but the answer is just 1/2 (a+b)? why would you not consider the 1/2 (a-b)? (also I can't type the mod signs so have just ignored them for now on here)

To type the modulus sign, it's "Shift + backslah (not the question mark one)". That's what's on my keyboard anyway...



Hmm... couple things.
1, We don't actually care where p is. After all, p will be a bunch of points. We are looking at the center of the circle formed by the p's.
2, Again, if you are saying p = 1/2 (a-b) + 1/2(a+b), p will just be one point - it doesn't form a circle.
3, Even worse, if you are saying p = 1/2 |a-b| + 1/2(a+b), that's nonsense since you are adding a scalar with a vector.

Sorry if this reply doesn't quite help. I'm not sure how to explain it other than just repeating "modulus means distance"...
Maybe someone else can chime in.

ohh thank you, I think I get it. so is it saying p varies around the centre 1/2(a+b) and 1/2(a-b) (modulus) will always be the distance between p and 1/2(a+b)?

Bingo!