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Section Formula - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

Section Formula is considered one of the most asked concept.
6 Questions around this concept.

Solve by difficulty

If the vectors $\underset{AB}{\rightarrow}$ $= 3\hat{i}+4\hat{k}$ and $\underset{AC}{\rightarrow}$ $= 5\hat{i}-2\hat{j}+4\hat{k}$ are the sides of a triangle ABC ,then the length of the median through A is :

A

$\sqrt{45}$

B

$\sqrt{18}$

C

$\sqrt{72}$

D

$\sqrt{33}$

Concepts Covered - 1

Section Formula

Let A and B be two points represented by the position vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ , respectively, with respect to the origin O.

Let R be a point that divides the line segment joining the points A and B in the ratio m : n.

Internal Division

If R divides AB internally in the ratio m : n, then position vector of R is given by $\overrightarrow{\mathrm{OR}}=\frac{m \vec{b}+n \vec{a}}{m+n}$

Proof :

$\\\text {Let } O \text { be the origin. Then } \overrightarrow{\mathrm{O} \mathrm{A}}=\mathrm{\mathbf{\vec a}} \text { and } \overrightarrow{\mathrm{OB}}=\mathrm{\mathbf{\vec b}} \text { . Let } \mathbf{\vec r} \text { be the position vector of R}\\\text{which divides AB internally in the ratio m : n. Then}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{AR}{RB}=\frac{m}{n}}\\\\\mathrm{or\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;n( \overrightarrow{AR} )=m( \overrightarrow{RB} )}\\\\\text{Now from triangles ORB and OAR, we have}$

$\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\overrightarrow{\mathrm{RB}}=\overrightarrow{\mathrm{OB}}-\overrightarrow{\mathrm{OR}}=\vec{b}-\vec{r}}\\\\\mathrm{and,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\overrightarrow{\mathrm{AR}}=\overrightarrow{\mathrm{OR}}-\overrightarrow{\mathrm{OA}}=\vec{r}-\vec{a}}\\\text{Therefore, we have}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;m(\vec{b}-\vec{r})=n(\vec{r}-\vec{a})}\\\\\mathrm{or\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\vec{r}=\frac{m \vec{b}+n \vec{a}}{m+n}$

Hence, the position vector of the point R which divides A and B internally in the ratio of m : n is given by

$\overrightarrow{\mathrm{OR}}=\frac{m \vec{b}+n \vec{a}}{m+n}$

External Division

If R divides AB externally in the ratio m : n, then position vector of R is given by $\overrightarrow{\mathrm{OR}}=\frac{m \vec{b}-n \vec{a}}{m-n}$

NOTE:

If R is the midpoint of AB, then m = n. And therefore, the midpoint R of $\overrightarrow{AB}$ , will have its position vector as $\overrightarrow{\mathrm{OR}}=\frac{\vec{a}+\vec{b}}{2}$

Study it with Videos

Section Formula

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