Q.Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $(2\vec{a} + \vec{b})$ and $(\vec{a} - 3\vec{b})$ externally in the ratio $1 : 2$. Also, show that P is the mid point of the line segment RQ.

Concept: Section Formula (External Division)

For two points with position vectors $\vec{p}$ and $\vec{q}$, the point dividing the segment externally in the ratio $m:n$ is given by:

Step 1 – Identify the vectors and ratio

Here $\vec{p} = 2\vec{a} + \vec{b}$, $\vec{q} = \vec{a} - 3\vec{b}$, and $m:n = 1:2$ (external).

Step 2 – Apply the external division formula

Step 3 – Simplify

$\vec{r} = 3\vec{a} + 5\vec{b}$ …

Concept and Intuition

The section formula is the backbone of this problem. When a point divides a line segment in a given ratio, we have two distinct cases:

• Internal division: The point lies between the two endpoints.
• External division: The point lies beyond one of the endpoints, on the line extended.

For external division, the formula looks almost like the internal one — but with a crucial sign change in the denominator. Why? Because when a point divides a segment externally, the distances are measured in opposite directions along the line, so one ratio component effectively becomes negative.

The second part of the problem asks us to show that P is the midpoint of RQ. This is a verification: once we have $\vec{r}$, we check whether $\vec{p} = \frac{\vec{r} + \vec{q}}{2}$.

Step-by-Step Solution

1. Identify the given vectors and ratio

We have:

• Position vector of P: $\vec{p} = 2\vec{a} + \vec{b}$
• Position vector of Q: $\vec{q} = \vec{a} - 3\vec{b}$
• Ratio: $1 : 2$ externally, with R dividing PQ. So $m = 1$, $n = 2$.

2. Apply the external section formula

Using $\vec{r} = \frac{m\vec{q} - n\vec{p}}{m - n}$:

3. Simplify the numerator

First, expand:

4. Divide by the denominator

Denominator is $1 - 2 = -1$. So:

$\vec{r} = \frac{-3\vec{a} - 5\vec{b}}{-1} = 3\vec{a} + 5\vec{b}$ …

Method: External Division by the Section Formula

Use this to find the point dividing a segment externally in a given ratio, then verify a midpoint relation if asked.

Steps

Step 1: Write the external section formula

If $R$ divides $PQ$ (position vectors $\vec p,\vec q$) externally in the ratio $m:n$, then

The minus signs (numerator and denominator) are what distinguish external from internal division.

Step 2: Substitute and simplify …

Common Mistakes

Mistake 1: Using the internal formula (plus signs) by mistake

Why it's wrong: internal division uses $\dfrac{m\vec q+n\vec p}{m+n}$; external needs the minus signs $\dfrac{m\vec q-n\vec p}{m-n}$. Correct approach: read 'externally' and switch to the minus form.

Mistake 2: Swapping which point gets weight $m$ versus $n$ …