question_answer

If a and b are the position vectors of A and B respectively, then the position vector of a point C on AB produced such that \[\overrightarrow{AC}=3\overrightarrow{AB}\] is                                                                                           [MNR 1980; MP PET 1995, 99]

A) \[3\mathbf{a}-\mathbf{b}\]

B) \[3\mathbf{b}-\mathbf{a}\]

C) \[3\mathbf{a}-2\mathbf{b}\]              

D) \[3\mathbf{b}-2\mathbf{a}\]

Correct Answer: D

Solution :

Since given that \[\overrightarrow{AC}=3\overrightarrow{AB}.\] It means that point \[C\] divides \[AB\] externally. Thus \[\overrightarrow{AC}:\overrightarrow{BC}=3:2\] Hence \[\overrightarrow{OC}=\frac{3.\mathbf{b}-2.\mathbf{a}}{3-2}=3\mathbf{b}-2\mathbf{a}.\]