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. Let us see in the upcoming discussion how section formula can be applied 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. 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:
\(\)\(\vec{OR}\)\(\) = \(\)\(\frac{\vec{b} + \vec{a}}{2}\)\(\)
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