find its coordinates?

Question

a pt on the segment whose end pts are (-1,3) and (5,2) is twice as far from the pt(-1,3) as from (5,2). Find its coordinates

please use vector equations

the answer is (3,7/3) but i don't nkow how to get that answer

Answer 1

Given A(-1,3) and B(5,2) find M(x,y) on AB such that M is twice as far from A as from B.

Let vector v = AB = <5+1,2-3> = <6,-1>

AM = 2MB

AB = AM + MB = AM + (1/2)AM
AB = (3/2)AM

AM = (2/3)AB = A + (2/3)v = (-1,3) + (2/3)<6,-1>
M = (-1+12/3,3-2/3) = (3,7/3)