Appeared in: 2024 (6 sets)
The position vectors of points P and Q are $\vec{p}$ and $\vec{q}$ respectively. The point R divides line segment PQ in the ratio $3:1$ and S is the mid-point of line segment PR. The position vector of S is : (A) $\frac{\vec{p} + 3\vec{q}}{4}$ (B) $\frac{\vec{p} + 3\vec{q}}{8}$ (C) $\frac{5\vec{p} + 3\vec{q}}{4}$ (D) $\frac{5\vec{p} + 3\vec{q}}{8}$
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Official Evaluation Steps
Step 1 (Formula / Governing Law): (½ Mark)
The position vector of a point dividing a line segment joining points with position vectors $\vec{a}$ and $\vec{b}$ in the ratio $m:n$ is given by the section formula:
$$\vec{r} = \frac{m\vec{b} + n\vec{a}}{m+n}$$The position vector of the mid-point of a line segment joining points with position vectors $\vec{a}$ and $\vec{b}$ is given by:
$$\vec{m} = \frac{\vec{a} + \vec{b}}{2}$$Step 2 (Step-by-Step Working): (½ Mark)
- Given position vectors of P and Q are $\vec{p}$ and $\vec{q}$.
- Point R divides PQ in the ratio $3:1$. Using the section formula: $$\vec{r} = \frac{3\vec{q} + 1\vec{p}}{3+1} = \frac{\vec{p} + 3\vec{q}}{4}$$
- S is the mid-point of PR. Using the mid-point formula with position vectors $\vec{p}$ and $\vec{r}$: $$\vec{s} = \frac{\vec{p} + \vec{r}}{2}$$
- Substituting the value of $\vec{r}$: $$\vec{s} = \frac{\vec{p} + \left( \frac{\vec{p} + 3\vec{q}}{4} \right)}{2}$$ $$\vec{s} = \frac{\frac{4\vec{p} + \vec{p} + 3\vec{q}}{4}}{2} = \frac{5\vec{p} + 3\vec{q}}{8}$$
Step 3 (Final Answer):
The position vector of S is (D) $\frac{5\vec{p} + 3\vec{q}}{8}$.