The physical quantities that have magnitude and are directly 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. Let us see in the upcoming discussion how we can apply section formula to vectors. 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. To locate the position of a point in space, we require a coordinate system.
Section Formula
If O is taken as reference origin and A is an arbitrary point in space then the vector
\(\begin{array}{l}\vec{OA}\end{array} \)
 is called the position vector of the point. Let us consider two points P and Q denoted by position vectors
\(\begin{array}{l}\vec{OP}\end{array} \)
 and
\(\begin{array}{l}\vec{OQ}\end{array} \)
  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 internally
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,
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:
I want section formula