Question

If \( \mathbf{a} \) and \( \mathbf{b} \) are position vectors of two points \( P \) and \( Q \) respectively, then find the position vector of a point \( R \) in \( QP \) produced such that \[ QR = \frac{3}{2} QP. \]

Hint:

If a point divides a line segment externally in the ratio \( m:n \), its position vector is given by: \[ \mathbf{r} = \frac{m\mathbf{b} - n\mathbf{a}}{m - n}. \]

Answer 1

Step 1: Define position vectors.
Let the position vectors of \( P \) and \( Q \) be: \[ \mathbf{OP} = \mathbf{a}, \quad \mathbf{OQ} = \mathbf{b}. \] The vector \( QP \) is given by: \[ \mathbf{QP} = \mathbf{a} - \mathbf{b}. \] Step 2: Express \( QR \) in terms of \( QP \).
Given that: \[ QR = \frac{3}{2} QP, \] we write: \[ \mathbf{QR} = \frac{3}{2} (\mathbf{a} - \mathbf{b}). \] Step 3: Compute the position vector of \( R \).
Using the relation: \[ \mathbf{OR} = \mathbf{OQ} + \mathbf{QR}, \] \[ \mathbf{OR} = \mathbf{b} + \frac{3}{2} (\mathbf{a} - \mathbf{b}). \] Expanding: \[ \mathbf{OR} = \mathbf{b} + \frac{3}{2} \mathbf{a} - \frac{3}{2} \mathbf{b}. \] \[ \mathbf{OR} = \frac{3}{2} \mathbf{a} + \left(1 - \frac{3}{2} \right) \mathbf{b}. \] \[ \mathbf{OR} = \frac{3}{2} \mathbf{a} - \frac{1}{2} \mathbf{b}. \] Final Answer: \[ \mathbf{r} = \frac{3}{2} \mathbf{a} - \frac{1}{2} \mathbf{b}. \]