(i) Let a, b be the position vectors of the given points A, B, with reference to any origin, O.
If r be the position vector of any point P on the locus, we have
PA2=PB2
⇒(r−a)2=(r−b)2
⇒−2r.a+a2=−2r.b+b2
⇒r.(a−b)=12(a2−b2)=12(a+b)(a−b)
⇒[r−12(a+b)]⋅(a−b)=0
Thus, the required locus is the plane bisecting the line AB normally.