Question

If \[\overrightarrow{a}\] and \[\overrightarrow{b}\] denote the position vectors of points A and B respectively and C is a point on AB such that 3AC = 2AB, then write the position vector of C.

Answer 1

Given: \[\overrightarrow{a}\] and \[\overrightarrow{b}\] are the position vectors of points A and B respectively and C  is a point on AB such that \[3 AC = 2AB .\] 
Let \[\vec{c}\] is the position vector of C 
Now,
\[\overrightarrow{AB} = \overrightarrow{b} - \overrightarrow{a}\]
\[\overrightarrow{AC} = \overrightarrow{c} - \overrightarrow{a}\]
Consider, 
\[3 AC \hspace{0.167em} = 2AB\]
\[ \Rightarrow 3 \left( \overrightarrow{c} - \overrightarrow{a} \right) = 2 \left( \overrightarrow{b} - \overrightarrow{a} \right)\]
\[ \Rightarrow 3 \overrightarrow{c} - 3 \overrightarrow{a} = 2 \overrightarrow{b} - 2 \overrightarrow{a} \]
\[ \Rightarrow 3 \overrightarrow{c} = 2 \overrightarrow{b} + \overrightarrow{a} \]
\[ \Rightarrow \overrightarrow{c} = \frac{1}{3} \left( 2 \overrightarrow{b} + \overrightarrow{a} \right)\]
\[ \Rightarrow \overrightarrow{c} = \frac{1}{3} \left( \overrightarrow{a} + 2 \overrightarrow{b} \right)\]
Hence, the position vector of C is \[\frac{1}{3}\left( \overrightarrow{a} + 2 \overrightarrow{b} \right)\]