Question

If ` A (bara) and B (barb) ` are any two points in the space
and ` R(barr)` be a point on the line segment AB
dividing it internally in the ratio m : n , then prove that :
`barr = (mbarb + nbara)/(m + n)`

Answer 1

As R is point on line segment AB (A - R - B ) and
` (AR)/(RB) = (m)/(n)`

` therefore m (RB) = n (AR) and bar(AR) and bar(AB) ` are in same direction .
` therefore m ((bar(RB)) = n (bar(AR))`
` rArr m (bar(OB) - bar(OR)) = n (bar(OR) - bar(OA)) `
` rArr m (barb - barr) = n (barr - bara)`
`rArr mbarb- mbarr = nbarr - nbara `
`rArr mbarb+ nbara = mbarr + n^(2)`
` rArr barr = (mbarb + nbara)/(m + n)` Hence Proved