question_answer

A and B are two points. The position vector of A is \[6\mathbf{b}-2\mathbf{a}.\] A point P divides the line AB in the ratio 1 : 2. If \[\mathbf{a}-\mathbf{b}\] is the position vector of P, then the position vector of B is given by                                                         [MP PET 1993]

A) \[7\mathbf{a}-15\mathbf{b}\]

B) \[7\mathbf{a}+15\mathbf{b}\]

C) \[2\pi /3\]

D) \[15\mathbf{a}+7\mathbf{b}\]

Correct Answer: A

Solution :

\[\overrightarrow{OP}=\frac{1(\overrightarrow{OB})+2(6\mathbf{b}-2\mathbf{a})}{1+2}\]             \[\Rightarrow 3(\mathbf{a}-\mathbf{b})=\overrightarrow{OB}+12\mathbf{b}-4\mathbf{a}\]\[\Rightarrow \overrightarrow{OB}=7\mathbf{a}-15\mathbf{b}\].