The position vectors of two points A and B are $\overrightarrow{\mathrm{OA}}=2\widehat{\mathrm{ı}}-\widehat{\mathrm{ȷ}}-\widehat{\mathrm{k}}\text{ and }\overrightarrow{\mathrm{OB}}=2\widehat{\mathrm{ı}}-\widehat{\mathrm{ȷ}}+2\widehat{\mathrm{k}}$ respectively. The position vector of a point P which divides the line segment joining A and B in the ratio 2 : 1 is ______.

$\overrightarrow{\mathrm{OP}}=2\stackrel{\wedge }{\mathrm{ı}}-\widehat{\mathrm{ȷ}}+\widehat{\mathrm{k}}$
Given: $\overrightarrow{\mathrm{OA}}=2\stackrel{\wedge }{\mathrm{ı}}-\widehat{\mathrm{ȷ}}-\widehat{\mathrm{k}},\overrightarrow{\mathrm{OB}}=2\widehat{\mathrm{ı}}-\widehat{\mathrm{ȷ}}+2\widehat{\mathrm{k}},\text{ Ratio }=2:1$ 
Now the poison vector of a point P which divides the line segment joining A and B in the ration 2 : 1 s given by, $\left(\frac{\mathrm{m}\overrightarrow{\mathrm{B}}+\mathrm{n}\overrightarrow{\mathrm{A}}}{\mathrm{m}+\mathrm{n}}\right)$
Hence,
$\overrightarrow{\mathrm{OP}}=\frac{2(2\hat{\mathrm{i}}-\hat{\mathrm{j}}+2\hat{\mathrm{k}})+1(2\hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}})}{2+1}=\frac{4\hat{\mathrm{i}}-2\hat{\mathrm{j}}+4\hat{\mathrm{k}}+2\hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}}}{3}=\frac{6\hat{\mathrm{i}}-3\hat{\mathrm{j}}+3\hat{\mathrm{k}}}{3}=2\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$
The position vector of a point P which divides the line segment joining A and B in the ratio 2 : 1 is $\overrightarrow{\mathrm{OP}}=2\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$