Section Formula in Vector Algebra - Definition, Formulas, Proof and Examples

What is Section Formula in Vector Algebra?

The Section Formula in Vector Algebra is used to find the position vector or coordinates of a point that divides the line joining two given vectors in a specified ratio. In simple words, it helps us determine how a line segment formed by two vectors is divided by another vector. This concept is widely used in solving problems related to position vectors, proportional division, midpoint, and vector geometry.

Let $A$ and $B$ be two points whose position vectors with respect to the origin $O$ are
$\overrightarrow{OA}=\vec{a}$ and $\overrightarrow{OB}=\vec{b}$ respectively.

Let $R$ be a point that divides the line segment $AB$ in the ratio $m:n$.
Then $R$ represents a new position vector that is a weighted combination of $\vec{a}$ and $\vec{b}$.

Types of Section Formula in Vector Algebra

The section formula in vector algebra is mainly divided into two types:

• Internal Section Formula

• External Section Formula

Internal Section Formula in Vector Algebra

Internal division means that the point dividing the line segment lies between the two given vectors. In other words, the dividing vector is located on the line joining the two vectors and inside the segment.

If point $R$ divides the line segment joining $A$ and $B$ internally in the ratio $m:n$, then the position vector of point $R$ is given by:

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

This formula is known as the Internal Section Formula in Vector Algebra. It is one of the most important results and is used to find midpoints, trisection points, and proportional division of vectors.

The External Section Formula is used when a point divides the line segment joining two vectors externally in a given ratio. In this case, the dividing point lies outside the line segment but still maintains a fixed proportional distance from the two given points. This concept is very important in advanced problems of vector geometry and coordinate-based vector algebra.

If point $R$ divides the line segment $AB$ externally in the ratio $m:n$, then the position vector of $R$ is given by:

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

This is known as the External Section Formula in Vector Algebra. It helps in locating points that lie beyond the given vectors while preserving the given ratio.

Midpoint Formula in Vector Algebra

The Midpoint Formula is a special case of the section formula where the line segment is divided into two equal parts. Here, the ratio of division is $1:1$, which means the point lies exactly halfway between the two given points.

A midpoint is the point that divides a line segment into two congruent segments.

If $R$ is the midpoint of the line segment $AB$, then
$m = n$

So, the position vector of the midpoint $R$ is:

$\overrightarrow{OR}=\frac{\vec{a}+\vec{b}}{2}$

This formula is widely used in Vector Algebra for finding the center of a line segment, checking symmetry, and simplifying vector-based constructions.

Solved Examples Based on Section Formula

Example 1: If the vectors $=3 \hat{i}+4 \hat{k}$ and $\overrightarrow{A C}=5 \hat{i}-2 \hat{j}+4 \hat{k}$ are the sides of a triangle ABC, then the length of the median through A is:
Solution: Midpoint formula - $\frac{\vec{a}+\vec{b}}{2}$
If $\vec{a}$ and, position vector of the mid-point of AB

$
\frac{\overrightarrow{A B}+\overrightarrow{A C}}{2}=\overrightarrow{A D}
$
$
\begin{aligned}
& \overrightarrow{A D}=4 \vec{i}+\vec{j}+4 \vec{k} \\
& |\overrightarrow{A D}|=\sqrt{4^2+1^2+4^2}=\sqrt{33}
\end{aligned}
$
Hence, the answer is $\sqrt{33}$

Example 2: If $C$ is the midpoint of $A B$ and $P$ is any point outside $A B$, then
Solution: If $\vec{a}$ and $\vec{b}$, position vector of mid-point of $A B$

$
\begin{aligned}
& \overrightarrow{P A}+\overrightarrow{P B}=(-\vec{P}+\vec{a})+(-\vec{P}+\vec{b}) \\
= & -2 \vec{P}+(\vec{a}+\vec{b}) \\
\overrightarrow{P C}= & \frac{\vec{a}+\vec{b}-2 \vec{P}}{2} \\
= & \frac{\overrightarrow{P A}+\overrightarrow{P B}}{2} \\
= & \overrightarrow{P A}+\overrightarrow{P B}=2 \overrightarrow{P C}
\end{aligned}
$
Hence, the answer is $\overrightarrow{P A}+\overrightarrow{P B}=2 \overrightarrow{P C}$

Example 3: The position vector of the point, which divides the join of the points having position vectors $\hat{i}+2 \hat{j}+\hat{k}$ and $-\hat{i}-\hat{j}+2 \hat{k}$ internally in ratio $2: 1$ is:
Solution: Let $\vec{a}=\hat{i}+2 \hat{j}+\hat{k}, \vec{b}=-\hat{i}-\hat{j}+2 \hat{k}$
And $m: n=2: 1$

$
\therefore \vec{r}=\frac{1(\hat{i}+2 \hat{j}+\hat{k})+2(-\hat{i}-\hat{j}+2 \hat{k})}{1+2}=\frac{-\hat{i}+5 \hat{k}}{3}
$
Hence, the answer is $\frac{1}{3}(-\hat{i}+5 \hat{k})$

Example 4: The vectors $\overrightarrow{A B}=3 \hat{i}+4 \hat{k}$ and $\overrightarrow{A C}=5 \hat{i}-2 \hat{j}+4 \hat{k}$ are the sides of a triangle $A B C$. The length of the median through $A$ is
Solution

$
\begin{aligned}
& \text { Median is } \overrightarrow{A D}=\frac{\overrightarrow{A B}+\overrightarrow{A C}}{2} \\
& =\frac{3 \hat{i}+4 \hat{k}+5 \hat{i}-2 \hat{j}+4 \hat{k}}{2} \\
& =4 \hat{i}-\hat{j}+4 \hat{k}
\end{aligned}
$
Length of Median $=|\overrightarrow{A D}|=\sqrt{4^2+1^2+4^2}=\sqrt{33}$
Hence, the answer is $\sqrt{33}$

List of Topics Related to Vector Algebra

This section gives a quick and clear overview of all the important topics in Vector Algebra, helping you understand what concepts are covered and how they are connected. It works like a roadmap for easy and organized learning.

Types of Vectors

Vectors and Scalars

Addition of Vectors and Subtraction of Vectors

Multiplication Of Vectors by a Scalar Quantity

Components Of A Vector Along And Perpendicular To Another Vector

NCERT Resources

This section provides carefully selected NCERT-based resources, including notes and solutions, to help you study exactly as per the syllabus and strengthen your basic concepts in Vector Algebra.

NCERT Maths Class 12th Notes for Chapter 10 - Vector Algebra

NCERT Maths Class 12th Solutions for Chapter 10 - Vector Algebra

NCERT Maths Class 12th Exemplar Solutions for Chapter 10 - Vector Algebra

Practice Questions based on Vector Algebra

This section offers well-chosen practice questions to help you apply Vector Algebra concepts, improve accuracy, and build confidence through regular problem solving.

Section Formula- Practice Question MCQ

We have provided below the practice questions based on the topics related to section formula: