Question

Let `A(bara)` and `B(barb)` 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 `barr = (mbarb + nbara)/(m + n)`.

If `A(bara)` and `B(barb)` 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 `barr = (mbarb + nbara)/(m + n)`.

Answer 1

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

∴ `n(vecr - veca) = m(vecb - vecr)`

∴ `nvecr - nveca = mvecb - mvecr`

∴ `mvecr + nvecr = mvecb + nveca`

∴ `(m + n)vecr = mvecb + nveca`

∴ `vecr = (mvecb + nveca)/(m + n)`