Question

Let $O$ be the origin. Let $\overrightarrow{OA} = \vec{a}$ and $\overrightarrow{OB} = \vec{b}$ be the position vectors of the points $A$ and $B$ respectively. A point $P$ divides the line segment $AB$ internally in the ratio $m:n$. Then $\overrightarrow{AP}$ is equal to:

• A. $\frac{2n(\vec{b}-\vec{a})}{m+n}$
• B. $\frac{n(\vec{b}+\vec{a})}{m+n}$
• C. $\frac{n(\vec{b}-\vec{a})}{m-n}$
• D. $\frac{m(\vec{b}-\vec{a})}{m+n}$
• E. $\frac{n(\vec{b}-\vec{a})}{m+n}$

Hint:

Always use $\vec{AP} = \vec{OP} - \vec{OA}$ after finding section formula.

Answer 1

The Correct Option is D

Concept:
• Position vector of division point: \[ \vec{OP} = \frac{m\vec{b} + n\vec{a}}{m+n} \]
• $\vec{AP} = \vec{OP} - \vec{OA}$

Step 1: Find $\vec{OP}$
\[ \vec{OP} = \frac{m\vec{b} + n\vec{a}}{m+n} \]

Step 2: Compute $\vec{AP}$
\[ \vec{AP} = \vec{OP} - \vec{OA} = \frac{m\vec{b} + n\vec{a}}{m+n} - \vec{a} \] \[ = \frac{m\vec{b} + n\vec{a} - (m+n)\vec{a}}{m+n} \] \[ = \frac{m(\vec{b} - \vec{a})}{m+n} \] Final Conclusion:
Option (D)