# 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:

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