Question

Let `A(bara)` and `B(barb)` be any two points in the space and `R(barr)` be the third point on the line AB dividing the segment AB externally in the ratio m : n, then prove that `barr = (mbarb - nbara)/(m - n)`.

Answer 1

As the point R divides the line segment AB externally, we have either

A – B – R or R – A – B

We assume that

A – B – R and `bar(AR) : bar(BR)` = m : n

∴ `(AR)/(BR) = m/n`,

.i.e. `n * l(AR) = m * l(BR)`

As `n(bar(AR))` and `m(bar(BR))` have

Same magnitude and direction,

∴ `n(bar(AR)) = m(bar(BR))`

∴ `n(barr - bara) = m(barr - barb)`

∴ `nbarr - nbara = mbarr - mbarb`

∴ `mbarr - nbarr = mbarb - nbara`

∴ `(m - n)barr = mbarb - nbara`

∴ `barr = (mbarb - nbara)/(m - n)`,

Hence proved.