Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are `hat i + 2 hat j - hat k` and `- hat i + hat j + hat k` respectively, in the ratio 2 : 1
(i) internally (ii) externally

Explanation

The position vector of point R dividing the line segement joining two points P and Q in the ratio `m : n` is given by
i. Internally : `(mvecb+nveca)/(m+n)` `" ""ii. Externally" : (mvecb-nveca)/(m-n)`
Position vectors of P and Q arer given as
`" "vec(OP)=hati+2hatj-hatk` and `vec(OQ)=-hati+hatj+hatk`
(i) The position vector of point R which divides the line joining two points P and Q internally in the ratio 2 : 1 is given by
`" "vec(OR)=(2(-hati+hatj+hatk)+1(hati+2hatj-hatk))/(2+1)=((-2hati+2hatj+2hatk)+(hati+2hatj-hatk))/(3)`
`" "=(-hati+4hatj+hatk)/(3)=-(1)/(3) hati+(4)/(3)hatj+(1)/(3)hatk`
(ii) The position vector of point R which divides the line joining two points P and Q externally in the ratio 2 : 1 is given by
`" "vec(OR)=(2(-hati+hatj+hatk)-1(hati+2hatj-hatk))/(2-1)=(-2hati+2hatj+2hatk)-(hati+2hatj-hatk)`
`" "=-3hati+3hatk`