Find the position vector of a point `R` which divides the line joining two points `P` and `Q` whose position vectors are `(2\ vec a+ vec b)` and (` vec a-\ 3 vec b)` respectively, externally in the ratio 1:2.Also, show that `P` is the mid-point of the line segment `R Qdot`

Explanation

Given that, `vec(OP) = 2 vec(a) + vec(b), vec(OQ) = vec(a) - 3vec(b)`
If a point R divides the line joining the points P and Q in the ratio m : n externally, then the position vector of R
m (position vector of point Q)
`=("-n(position vector of point P)")/(m-n)`
`therefore` Position vector of point R
`=((vec(a) - 3vec(b))xx1-(2vec(a) + vec(b))xx2)/(1-2)`
`(vec(a) - 3vec(b)-4vec(a)-2vec(b))/(-1)`
`=(-3vec(a)-5vec(b))/(-1) = 3vec(a)+vec(b)`
Now, the position vector of mid-point of RQ
`=(vec(OQ)+vec(OR))/(2) = ((3vec(a) + 5vec(b))+(vec(a) -3vec(b)))/(2)`
`=(4vec(a) + 2vec(b))/(2) = 2vec(a) + vec(b)`
which is the position vector of point P also implies P is the mid-point of line segment RQ.
Hence Proved.