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Question
Write the position vector of the point which divides the join of points with position vectors \[3 \overrightarrow{a} - 2 \overrightarrow{b}\text{ and }2 \overrightarrow{a} + 3 \overrightarrow{b}\] in the ratio 2 : 1.
Solution
Suppose R be the point which divides the line joining the points with position vectors \[3 \overrightarrow{a} - 2 \overrightarrow{b}\text{ and }2 \overrightarrow{a} + 3 \overrightarrow{b}\] in the ratio 2 : 1
And, \[\overrightarrow{OA} = 3 \overrightarrow{a} - 2 \overrightarrow{b}\text{ and }\overrightarrow{OB} = 2 \overrightarrow{a} + 3 \overrightarrow{b}\]
Here, m : n = 2 : 1
Therefore, position vector \[\overrightarrow{OR}\] is as follows:
\[\overrightarrow{OR} = \frac{m \overrightarrow{OB} + n \overrightarrow{OA}}{m + n}\]
\[ = \frac{2(2 \overrightarrow{a} + 3 \overrightarrow{b} ) + 1(3 \overrightarrow{a} - 2 \overrightarrow{b} )}{2 + 1}\]
\[ = \frac{7 \overrightarrow{a} + 4 \overrightarrow{b}}{3}\]
\[ = \frac{7}{3} \overrightarrow{a} + \frac{4}{3} \overrightarrow{b}\]
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