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. 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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