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

Here, $\vec a = \hat i + 2\hat j - \hat k$ and $\vec b = - \hat i + \hat j + \hat k$

(i) The position vector of $R$ dividing the join of $P$ and $Q$ internally in the ratio 2 :

1 is
$\vec R = \cfrac{{m\vec b + n\vec a}}{{m + n}} = \cfrac{{2(\vec b) + 1(\vec a)}}{{2 + 1}}$

$= \cfrac{{2( - \hat i + \hat j + \hat k) + 1(\hat i + 2\hat j - \hat k)}}{{2 + 1}} = - \cfrac{1}{3}\hat i + \cfrac{4}{3}\hat j + \cfrac{1}{3}\hat k$

(ii) The position vector of $R$ dividing the join of $P$ and $Q$ externally in the ratio 2 :

1 is
$\vec R = \cfrac{{m\vec b - n\vec a}}{{m - n}} = \cfrac{{2(\vec b) - 1(\vec a)}}{{2 - 1}} = \cfrac{{2( - \hat i + \hat j + \hat k) - 1(\hat i + 2\hat j - \hat k)}}{{2 - 1}}$
$= - 3\hat i + 0\hat j + 3\hat k = - 3\hat i + 3\hat k$