If $\vec{a}$ and $\vec{b}$ 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.

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Solution

Given: $\vec{a}$ and $\vec{b}$ are the position vectors of points $A$ and $B$ respectively and $C$ is a point on $A B$ such that $3 A C = 2 A B .$
Let $\vec{c}$ is the position vector of C
Now,
$\vec{A B} = \vec{b} - \vec{a}$
$\vec{A C} = \vec{c} - \vec{a}$
Consider,
$3 A C = 2 A B \Rightarrow 3 (\vec{c} - \vec{a}) = 2 (\vec{b} - \vec{a}) \Rightarrow 3 \vec{c} - 3 \vec{a} = 2 \vec{b} - 2 \vec{a} \Rightarrow 3 \vec{c} = 2 \vec{b} + \vec{a} \Rightarrow \vec{c} = \frac{1}{3} (2 \vec{b} + \vec{a}) \Rightarrow \vec{c} = \frac{1}{3} (\vec{a} + 2 \vec{b})$

Hence, the position vector of $C$ is $\frac{1}{3} (\vec{a} + 2 \vec{b})$

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