The position vector of the point which divides the join of points $2\mathbf{a} - 3\mathbf{b}$ and $\mathbf{a} + \mathbf{b}$ in the ratio $3 : 1$, is

$\frac{5\mathbf{a}}{4}$

The correct answer is Option (4) → $\frac{5\mathbf{a}}{4}$ ##

Since, the position vector of a point $R$ dividing the line segment joining the points $P$ and $Q$ whose position vectors are $\mathbf{p}$ and $\mathbf{q}$ in the ratio $m : n$ internally, is given by $\frac{m\mathbf{q} + n\mathbf{p}}{m + n}$.

Let $OP = 2\mathbf{a} - 3\mathbf{b}, OQ = \mathbf{a} + \mathbf{b}$

Let the position vector of the point $R$ divides the join of points $2\mathbf{a} - 3\mathbf{b}$ and $\mathbf{a} + \mathbf{b}$

$∴\text{Position vector } R = \frac{3(\mathbf{a} + \mathbf{b}) + 1(2\mathbf{a} - 3\mathbf{b})}{3 + 1}$

$R = \frac{5\mathbf{a}}{4}$