Ratios and coefficient comparison

For AP:PB=m:n with P between A and B, the displacement from A is AP= mm+n(b- a). OP= a+ mm+n(b- a)= n a+m bm+n. The coefficient nearer b belongs to the distance measured from A. Example: AP:PB=2:3 gives OP= 35 a+ 25 b. The coefficients add to 1, and both are between 0 and 1. If a and b are non-parallel, equate their coefficients in two expressions for the same vector. For example, p a+2 b=3 a+(q-1) b gives p=3,q=3. To show points are collinear, show that their connecting vectors are scalar multiples with a common point. A negative multiplier puts the point in the opposite direction; a multiplier greater than 1 can put it beyond the segment.