Section Formula in Vector Algebra

The section formula in vector algebra is used to find the position vector of a point that divides the line segment joining two given points in a fixed ratio. It connects vector representation, ratio, midpoint, centroid, and geometric interpretation in a simple algebraic form.

\[\mathbf{\overline{r}}=\mathbf{\frac{m\overline{b}+n\overline{a}}{m+n}}\]

\[\overline{\mathrm{r}}=\frac{\mathrm{m\overline{b}-n\overline{a}}}{\mathrm{m-n}}\]

If R (r̄) is the mid-point of the line segment joining the points A (ā) and B (b̄), then

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

Centroid of Triangle:

\[\mathbf{\overline{g}}=\frac{\mathbf{\overline{a}}+\mathbf{\overline{b}}+\mathbf{\overline{c}}}{3}\]

Centroid of Tetrahedron:

\[\overline{\mathbf{g}}=\frac{\overline{\mathbf{a}}+\overline{\mathbf{b}}+\overline{\mathbf{c}}+\overline{\mathbf{d}}}{4}\]

Incentre of Triangle:

\[\overline{\mathrm{h}}=\frac{\left|\overline{\mathrm{AB}}\right|\overline{\mathrm{c}}+\left|\overline{\mathrm{BC}}\right|\overline{\mathrm{a}}+\left|\overline{\mathrm{AC}}\right|\overline{\mathrm{b}}}{\left|\overline{\mathrm{AB}}\right|+\left|\overline{\mathrm{BC}}\right|+\left|\overline{\mathrm{AC}}\right|}\]

Orthocentre of Triangle:

\[\overline{\mathrm{p}}=\frac{\tan A\left(\overline{\mathrm{a}}\right)+\tan B\left(\overline{\mathrm{b}}\right)+\tan C\left(\overline{\mathrm{c}}\right)}{\tan A+\tan B+\tan C}\]

Consider two points \[P\] and \[Q\] with position vectors \[\vec{OP} = 3\vec{a} - 2\vec{b}\] and \[\vec{OQ} = \vec{a} + \vec{b}\]. Find the position vector of a point \[R\] which divides the line joining \[P\] and \[Q\] in the ratio 2:1, (i) internally, and (ii) externally.

Solution:

1. The position vector of the point \[R\]dividing the join of \[P\] and \[Q\] internally in the ratio 2:1 is
   \[\vec{OR} = \frac{2(\vec{a} + \vec{b}) + (3\vec{a} - 2\vec{b})}{2 + 1} = \frac{5\vec{a}}{3}\]
2. The position vector of the point \[R\] dividing the join of \[P\] and \[Q\] externally in the ratio 2:1 is
   \[\vec{OR} = \frac{2(\vec{a} + \vec{b}) - (3\vec{a} - 2\vec{b})}{2 - 1} = 4\vec{b} - \vec{a}\]

• A point dividing a road between two cities in a fixed ratio can be represented mathematically using the same weighted-position idea as the section formula.

• In computer graphics and design, intermediate positions between two points are often estimated using ratio-based placement, which is closely related to the idea behind section formula.

• Section formula gives the position vector of a point dividing a line segment in a given ratio.

• For internal division, use \(\dfrac{m\vec{b}+n\vec{a}}{m+n}\).

• For external division, use \(\dfrac{m\vec{b}-n\vec{a}}{m-n}\).

• Midpoint is the special case when the ratio is \(1:1\).

• Centroid formulas are natural extensions of the same averaging idea.