A) \[3\mathbf{b}-\frac{\mathbf{a}}{2}\]
B) \[3\mathbf{b}+\frac{\mathbf{a}}{2}\]
C) \[3\mathbf{b}-\frac{\mathbf{a}}{3}\]
D) \[3\mathbf{b}+\frac{\mathbf{a}}{3}\]
Correct Answer: C
Solution :
Since \[\overrightarrow{OA}=\mathbf{a},\] \[\overrightarrow{OB}=\mathbf{b}\] and \[2AC=CO\] By section formula \[\overrightarrow{OC}=\frac{2}{3}\mathbf{a}.\] Therefore, \[|\overrightarrow{CD}|=3|\overrightarrow{OB}|\,\Rightarrow \overrightarrow{CD}=3\mathbf{b}\] \[\Rightarrow \overrightarrow{OD}=\overrightarrow{OC}+\overrightarrow{CD}=\frac{2}{3}\mathbf{a}+3\mathbf{b}\] Hence, \[\overrightarrow{AD}=\overrightarrow{OD}-\overrightarrow{OA}=\frac{2}{3}\mathbf{a}+3\mathbf{b}-\mathbf{a}=3\mathbf{b}-\frac{1}{3}\mathbf{a}.\]You need to login to perform this action.
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