Q.The position vector of the point which divides the join of points $2\vec{a}-3\vec{b}$ and $\vec{a}+\vec{b}$ in the ratio 3 : 1 is
(A) $\dfrac{3\vec{a}-2\vec{b}}{2}$
(B) $\dfrac{7\vec{a}-8\vec{b}}{4}$
(C) $\dfrac{3\vec{a}}{4}$
(D) $\dfrac{5\vec{a}}{4}$

Concept: Section Formula — For vectors, if a point $P$ divides the join of $A(\vec{p})$ and $B(\vec{q})$ in the ratio $m:n$ internally, then

$\vec{P} = \dfrac{n\vec{p} + m\vec{q}}{m+n}$.

Step 1: Identify the given vectors.

Let $\vec{A} = 2\vec{a} - 3\vec{b}$ and $\vec{B} = \vec{a} + \vec{b}$.

The ratio is $3:1$, so $m=3$, $n=1$ (from $A$ to $B$).

Step 2: Apply the section formula:

$\vec{P} = \dfrac{1 \cdot (2\vec{a} - 3\vec{b}) + 3 \cdot (\vec{a} + \vec{b})}{3+1}$. …

The section formula for vectors is the direct analogue of the coordinate geometry formula. If a point $P$ divides the line segment joining $A$ (with position vector $\vec{A}$) and $B$ (with position vector $\vec{B}$) in the ratio $m:n$, then the position vector of $P$ is given by:

This works because we are taking a weighted average of the endpoints, where the weight of each endpoint is proportional to the distance to the opposite division point. For internal division, the formula is symmetric and intuitive: the point closer to $B$ gets a larger weight from $B$, and vice versa.

Now, let’s apply this to the given problem.

1. Identify the endpoints and the ratio.

   Let $\vec{A} = 2\vec{a} - 3\vec{b}$ and $\vec{B} = \vec{a} + \vec{b}$.

   The ratio is $3:1$, meaning $m=3$ and $n=1$ (the point is closer to $B$ since $m > n$).

2. Plug into the section formula.

3. Simplify the numerator. …

Method: Internal Division by the Section Formula (Vector Form)

Use this to find the position vector of a point dividing a segment in a given ratio.

Steps

Step 1: Identify endpoints and match the ratio to weights

For a point dividing the join of $\vec{A}$ and $\vec{B}$ internally in the ratio $m:n$,

Note the cross-pairing: the far endpoint $\vec{B}$ carries weight $m$, the near endpoint $\vec{A}$ carries weight $n$.

Step 2: Substitute and simplify …

Common Mistakes

Mistake 1: Swapping the weights $m$ and $n$

Why it's wrong: putting $m$ on the near endpoint gives $\frac{3(2\vec{a}-3\vec{b})+1(\vec{a}+\vec{b})}{4}=\frac{7\vec{a}-8\vec{b}}{4}$ — the distractor option B. Correct approach: the far endpoint $\vec{B}$ takes weight $m$, the near endpoint $\vec{A}$ takes weight $n$: $\vec{r}=\frac{n\vec{A}+m\vec{B}}{m+n}$.

Mistake 2: Using the external-division formula

Why it's wrong: the denominator $m-n$ is for external division; this point divides internally. Correct approach: use $m+n$ in the denominator. …