# Section Formula in Vector Algebra

## Section Formula

The physical quantities which have magnitude, as well as direction attached to them, are known as vectors. Position vectors simply denote the position or location of a point in the three-dimensional Cartesian system with respect to a reference origin.

The concept of section formula is implemented to find the coordinates of a point dividing a line segment internally or externally in a specific ratio. In order to locate the position of a point in space, we require a coordinate system.

If O is taken as reference origin and A is an arbitrary point in space then the vector $\vec{OA}$  is called as the position vector of the point. Let us consider two points P and Q denoted by position vectors $\vec{OP}$  and $\vec{OQ}$  with respect to origin O.

Let us consider that the line segment connecting P and Q is divided by a point R lying on PQ. The point R can divide the line segment PQ in two ways: internally and externally. Let us consider both these cases individually.

#### Case 1: Line segment PQ is divided by R internallyf

Let us consider that the point R divides the line segment PQ in the ratio m: n, given that m and n are positive scalar quantities we can say that,

m$\overline{RQ}$ = n$\overline{PR}$

Consider the triangles, ∆ORQ and ∆OPR.

$\overline{RQ}$ = $\overline{OQ}$ – $\overline{OR}$ = $\vec{b}$ – $\vec{r}$ $\overline{PR}$ = $\overline{OR}$ – $\overline{OP}$ = $\vec{r}$ – $\vec{a}$

Therefore,

m($\vec{b}$ – $\vec{r}$) = n($\vec{r}$ – $\vec{a}$)

Rearranging this equation we get:

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

Therefore the position vector of point R dividing P and Q internally in the ratio m:n is given by:

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

#### Case 2: Line segment PQ is divided by R externally

Let us consider that the point R divides the line segment PQ in the ratio m: n, given that m and n are positive scalar quantities we can say that,

m$\overline{RQ}$ = -n$\overline{PR}$

Consider the triangles, ∆ORQ and ∆OPR.

$\overline{RQ}$ = $\overline{OQ}$ – $\overline{OR}$ = $\vec{b}$ – $\vec{r}$ $\overline{PR}$ = $\overline{OR}$ – $\overline{OP}$ = $\vec{r}$ – $\vec{a}$

Therefore,

m($\vec{b}$ – $\vec{r}$) = -n($\vec{r}$ – $\vec{a}$)

Rearranging this equation we get:

$\vec{r}$ = $\frac{m\vec{b} – n\vec{a}}{m – n}$

Therefore the position vector of point R dividing P and Q internally in the ratio m:n is given by:

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

What if the point R dividing the line segment joining points P and Q is the midpoint of line segment AB?

In that case, if R is the midpoint, then R divides the line segment PQ in the ratio 1:1, i.e. m = n = 1.The position vector of point R dividing will be given as:

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