Section Formula - Vector Algebra

The section formula is used to determine the position vector of a point that divides the line segment joining two given points in a specified ratio. In vector algebra, it provides a direct method for finding the coordinates or position vector of the dividing point without measuring distances geometrically.

Let the position vectors of points P and Q be and

Suppose a point R divides the line segment PQ in the ratio m:n in two ways as follows:

1. Internally

When R divides PQ internally. If R divides such that

where m and n are positive values, we specify that the point R divides  internally in the ratio of m : n. 

👁 1

Now from triangles ORQ and OPR, we have

Hence, we can conclude that,

m  = n 

On simplification, we get

Midpoint Formula : When the point divides the line segment equally, then m = n

So, we get

which is called the midpoint formula.

2. Externally

When R divides PQ externally. If R divides  such that

where m and n are positive values, we say that the point R divides  externally in the ratio of the m : n, provided m ≠ n.

👁 2

Now from triangles ORQ and OPR, we have

Hence, we can conclude that,

On simplification, we get

Solved Examples

Problem 1: Find the position vectors of the points which divide the join of the points   and  internally and externally in the ratio 2 : 3.

Let A and B be the given points with the position vectors  and   respectively.

Let P divide the in the ratio 2 : 3 internally

m = 2 and n = 3

Using internally section formula,

👁 3

Position vector of P = 

Position vector of P = 

Position vector of P = 

Position vector of P = 

Now, Let P divide the  in the ratio 2 : 3 externally

m = 2 and n = 3

Using externally section formula,

👁 4

Position vector of P = 

Position vector of P = 

Position vector of P = 

Position vector of P = 

Problem 2: If  and  are position vectors of points A and B respectively, then find the position vector of points of trisection of AB.

Let P and Q be points of trisection. Then, AP = PQ = QB = k (constant variable)

PB = PQ + QB = k + k = 2k

P divides AB in the ratio 1 : 2

👁 5

Using internally section formula, where m=1 and n=2

Position vector of P = 

Position vector of P = 

Position vector of P = 

Now, we can clearly see that Q is the mid-point of PB.

Apply mid-point section formula we have,

Position vector of Q = 

Position vector of Q = 

Position vector of Q = 

Practice Problems

1. Find the midpoint of points having position vectors
2. Find the position vector of a point dividing the join of a and b internally in the ratio 3:2.
3. Find the point dividing the segment joining externally in the ratio 2:1.
4. Determine the trisection points of the segment joining position vectors
5. Show that the midpoint of the line segment joining points with position vectors and is the origin.