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

Explanation

The position vector of a point R divide the line segment joining two points P and Q in the ratio m:n is given by
Case I Internally `=(mvec(b)+nvec(a))/(m+n)`
Case II Externally `=(mvec(b)-nvec(a))/(m-n)`
Position vectors of P and Q are given as `OP=hat(i)+2hat(j) and OQ=-hat(i)+hat(j)+hat(k)`
(i) PV of R [dividing (PQ) in the ration `m:n` internally]
`=(m(PV of Q)+n(PV of P))/(m+n)`
Here `m=2,n=1=(2(PV of Q)+1(PV of P))/(2+1)`
`=(1)/(3){2(-hat(i)+hat(j)+hat(k))+1(hat(i)+2hat(j)-hat(k))}`
`=(1)/(3)(-hat(i)+4hat(j)+hat(k))=(-(1)/(3))hat(i)+((4)/(3))hat(j)+((1)/(3))hat(k)`
(ii) PV of R [dividing (PQ)in the ratio `2:1` externally]
`=(m(PV of Q)-n(PV of P))/(m-n)`
Here `m=2,n=1=(2(PV of Q)-1(PV of P))/(2-1)`
`=2(-hat(i)+hat(j)+hat(k))+(-hat(1))(hat(i)+2hat(j)-hat(k))=(-3)hat(i)+0hat(j)+3hat(k)=-3hat(i)+3hat(k)`.