Section Formula

The physical quantities that have the magnitude, as well as the direction attached to them, are called as vectors. Position vectors denote the location or the position of a point in three-dimensional Cartesian system with respect to an origin. Let us take a look in the upcoming discussion on how you can apply the section formula in vectors. The concept of section formula is implemented for finding the coordinates of a point dividing a line segment either internally or externally in a particular ratio. For locating the position of a point in space, you need a coordinate system.

Consider the figure given below:

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Here, P and Q are the points that are represented through the position vectors \[\overrightarrow{OP}\] and \[\overrightarrow{OQ}\] respectively, with respect to the origin 0. You can divide the line segment that divides the two points P and Q with the help of a third point R in two different points: internally and externally. Let us take a look at booth these cases individually.

Section Formula In Coordinate Geometry

If you wish to find the position vector \[\overrightarrow{OQ}\] consider the following cases one by one.

Case 1: When R divides the segment PQ internally

Take a look at this figure again.

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From the figure, the point R divides \[\overrightarrow{PQ}\] in a way that

m \[\overrightarrow{RQ}\] - n \[\overrightarrow{PR}\] ... (1)

Here, m and n are called as positive scalars and you can say that the point R divides 

\[\overrightarrow{PQ}\] internally in m:n ratio. 

Now, take a look at the triangles ORQ and OPR. You have

\[\overrightarrow{RQ}\] = \[\overrightarrow{OQ}\] - \[\overrightarrow{OR}\] = \[\overrightarrow{b}\]  - \[\overrightarrow{r}\] and

\[\overrightarrow{PR}\] = \[\overrightarrow{OR}\] -  \[\overrightarrow{OP}\] = \[\overrightarrow{r}\] - \[\overrightarrow{a}\]

When you replace the values of \[\overrightarrow{RQ}\] and \[\overrightarrow{PR}\] in the equation 1, you get,

m( \[\overrightarrow{b}\] - \[\overrightarrow{r}\] ) = n(\[\overrightarrow{r}\] - \[\overrightarrow{a})\]

or

\[\overrightarrow{r}\]  = \[\frac{\overrightarrow{mb}  +  \overrightarrow{na}}{m + n}\] ...(2)

Therefore, the position vector formula of point R that divides PQ internally in m:n ratio is given by

\[\overrightarrow{OR}\] = \[\frac{\overrightarrow{mb}  +  \overrightarrow{na}}{m + n}\] 

Case 2: When R divides the segment externally

Consider the figure given below:

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In the given figure, the point R divides PQ externally in the m:n ratio. Hence, you can say that the point Q divides PR internally in the (m - n): n ratio. 

Hence,

 \[\frac{PQ}{QR}\] = \[\frac{m - n }{n}\]

When you use the equation 2, you get,

 \[\overrightarrow{b}\] = \[\frac{(m - n)\overrightarrow{r} + \overrightarrow{na}}{(m - n) + n}\]

\[\overrightarrow{b}\] = \[\frac{(m - n)\overrightarrow{r} + \overrightarrow{na}}{m}\]

\[\overrightarrow{mb}\] = (m - n)\[\overrightarrow{r}\] + \[\overrightarrow{na}\]

You can also write this as

\[\overrightarrow{mb}\] - \[\overrightarrow{na}\] = (m - n) \[\overrightarrow{r}\]

Therefore, 

\[\overrightarrow{r}\] = \[\frac{\overrightarrow{mb} - \overrightarrow{na}}{m - n}\] ...(3)

When R is the midpoint of PQ, m = n. 

Hence, from equation 2 you have

\[\overrightarrow{OR}\] = \[\frac{\overrightarrow{mb} + \overrightarrow{na}}{m + n}\]

or,

\[\overrightarrow{OR}\] = \[\frac{m(\overrightarrow{b} + \overrightarrow{a})}{2m}\]

Therefore, 

\[\overrightarrow{r}\] = \[\frac{\overrightarrow{b} + \overrightarrow{a}}{2}\]

Therefore, the position vector formula of the midpoint R of PQ is given by,

\[\overrightarrow{OR}\] =  \[\frac{\overrightarrow{b} + \overrightarrow{a}}{2}\]

Section Formula Examples

Example 1:

Let two points P and Q have position vectors \[\overrightarrow{OP}\] = \[\overrightarrow{3a}\] - \[\overrightarrow{2b}\] and \[\overrightarrow{OQ}\] = \[\overrightarrow{a}\] + \[\overrightarrow{b}\]

Find the position vector formula of the point R that divides the line joining P and Q in the 2:1 ratio internally and externally.

Solution:

Since the point R divides PQ in the 2:1 ratio, you have m = 2, n = 1.

When R divides PQ internally.

From equation 2, you have

\[\overrightarrow{r}\] = \[\frac{\overrightarrow{mb} + \overrightarrow{na}}{m + n}\]

Hence,

\[\overrightarrow{r}\] = \[\frac{2(\overrightarrow{a} + \overrightarrow{b}) + (\overrightarrow{3a} - \overrightarrow{2b})}{2 + 1}\]  = \[\frac{\overrightarrow{5a}}{3}\]

When R divides PQ externally,

Considering equation 3,

\[\overrightarrow{r}\] = \[\frac{\overrightarrow{mb} - \overrightarrow{na}}{m - n}\]

Hence,

\[\overrightarrow{r}\] = \[\frac{2(\overrightarrow{a} + \overrightarrow{b}) - (\overrightarrow{3a} - \overrightarrow{2b})}{2-1}\] = \[\overrightarrow{4b}\] - \[\overrightarrow{a}\]

Example 2

Consider A=(−3, 1), B = (3, -6). Determine the coordinates of the point P (x, y) that divides the line segment AB internally in the ratio 1:2.

Solution:

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The given point P is \[\frac{1}{1 + 2}\] x AB far from the point A.

If you measure it parallel to the x-axis, you get,

x = -3 + \[\frac{1}{3}\] x (3 - (-3)) = -1

If you measure it parallel to the y-axis, you get,

y = 1 + \[\frac{1}{3}\] x (-6 -1) = \[\frac{-4}{3}\]

Hence, the coordinates of the point P are ( -1, \[\frac{-4}{3}\] )