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
The position vector of point $$R$$ dividing the line segment joining two points $$P$$ and $$Q$$ in the ratio $$m:n$$ is given by:
i. Internally:
$$\dfrac {m\vec {Q}+n\vec {P}}{m+n}$$
ii. Externally:
$$\dfrac {m\vec {Q}-n\vec {P}}{m-n}$$
Position vectors of $$P$$ and $$Q$$ are given as:
$$\vec {OP}=\hat {i}+2\hat {j}-\hat {k}$$ and $$\vec {OQ}=-\hat {i}+\hat {j}+\hat {k}$$
(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,
$$\vec {OR}=\dfrac {2(-\hat {i}+\hat {j}+\hat {k})+1(\hat {i}+2\hat {j}-\hat {k})}{2+1}=\dfrac {-2\hat {i}+2\hat {j}+\hat {k}+(\hat {i}+2\hat {j}-\hat {k})}{3}$$
$$=\dfrac {-\hat {i}+4\hat {j}+\hat {k}}{3}=-\dfrac {1}{3}\hat {i}+\dfrac {4}{3}\hat {j}+\dfrac {1}{3}\hat {k}$$
(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,
$$\vec {OR}=\dfrac {2(-\hat {i}+\hat {j}+\hat {k})-1(\hat {i}+2\hat {j}-\hat {k})}{2-1}=(-2\hat {i}+2\hat {j}+2\hat {k})-(\hat {i}+2\hat {j}-\hat {k})$$
$$=-3\hat {i}+3\hat {k}$$
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