Question

Find a position vector of a point \(R\) which devides 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 \(2\ratio 1\)
(i)internally
(ii)externally

Answer 1

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:
\(\frac{m\vec{b}+n\vec{a}}{m+n}\)
ii.Externally:
\(\frac{m\vec{b}-n\vec{a}}{m-n}\)
Position vectors of \(P\) and \(Q\) are given as:
\(\vec{OP}=\hat{i}+2\hat{j}-\hat{k}\) and \(\hat{i}+\hat{j}+\hat{k}\)
(i)The position vector of point \(R\) which divides the line joining two points \(P\) and \(Q\) internally in the ratio \(2\ratio 1\) is given by,
\(\vec{OR}=\frac{2(-\hat{i}+\hat{j}+\hat{k})+1(\hat{i}+2\hat{j}-\hat{k})}{2+1}\)
\(=\frac{(-2\hat{i}+2\hat{j}+2\hat{k})+(\hat{i}+2\hat{j}-\hat{k})}{3}\)
\(=\frac{-\hat{i}+4\hat{j}+\hat{k}}{3}\)
\(=\frac{-1}{3}\hat{i}+\frac{4}{3}\hat{j}+\frac{1}{3}\hat{k}\)
(ii)The position vector of point \(R\) which devides the line joining \(P\) and \(Q\) externally in the ratio \(2\ratio 1\) is given by,
\(\vec{OR}=\frac{2(-\hat{i}+\hat{j}+\hat{k})-1(\hat{i}+2\hat{j}-\hat{k})}{2-1}\)
\(=(-2\hat{i}+2\hat{j}+2\hat{k})-(\hat{i}+2\hat{j}-\hat{k})\)
\(=-3\hat{i}+3\hat{k}\).