A) \[\frac{\sqrt{136}}{9}\]
B) \[\frac{\sqrt{136}}{3}\]
C) \[\frac{20}{3}\]
D) \[\sqrt{\frac{217}{9}}\]
E) \[\frac{25}{3}\]
Correct Answer: B
Solution :
\[\because \] \[\overrightarrow{OA}=2\hat{i}+2\hat{j}+\hat{k}\] and \[\overrightarrow{OB}=2\hat{i}+4\hat{j}+4\hat{k}\] We have, \[|\overrightarrow{OA}|=\sqrt{4+4+1}=3\] and \[|\overrightarrow{OB}|=\sqrt{4+16+16}=6\] \[\therefore \] Required vector\[=\lambda (O\hat{A}+O\hat{B})\] \[=\lambda \left[ \frac{1}{3}(2\hat{i}+2\hat{j}+\hat{k})+\frac{1}{6}(2\hat{i}+4\hat{j}+4\hat{k}) \right]\] \[=\lambda \left[ \frac{1}{3}(2\hat{i}+2\hat{j}+\hat{k}+\hat{i}+2\hat{j}+2\hat{k}) \right]\] \[=\frac{\lambda }{3}(3\hat{i}+4\hat{j}+3\hat{k})\] \[\therefore \]Length of vector\[=\frac{\lambda }{3}\sqrt{9+16+9}=\frac{\lambda }{3}\sqrt{34}\] Take \[\lambda =2\] \[\therefore \]Required length of a vector is\[\frac{\sqrt{136}}{3}\]
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