Let `"A" (bar"a")` and `"B" (bar"b")` are any two points in the space and `"R"(bar"r")` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `bar "r" = ("m"bar"b" + "n"bar"a")/("m" + "n") `

#### Solution

R is a point on the line segment AB(A-R-B) and `bar"AR"` and `bar"RB"` are in the same direction.

Point R divides AB internally in the ratio m : n

∴ `"AR"/"RB" = "m"/"n"`

∴ n(AR) = m(RB)

As n `(bar"AR")` and `"m"(bar"RB")` have same direction and magnitude,

`"n"(bar"AR") = "m"(bar("RB"))`

∴ `"n"("OR" - bar"OA") = "m"(bar"OB" - bar"OR")`

∴ `"n"(bar"r" - bar"a") = "m"(bar"b" - bar"r")`

∴ `"n"bar"r" - "n"bar"a" = "m"bar"b" - "m"bar"r"`

∴ `"m"bar"r" + "n"bar"r" = "m"bar"b" + "n"bar"a"`

∴ `("m" + "n")bar"r" = "m"bar"b" + "n"bar"a"`

∴ `bar"r" = ("m"bar"b" + "n"bar"a")/("m" + "n")`