1. Meaning of the Section Formula
The section formula helps find the position vector of a point that divides a line segment joining two points in a specific ratio. Instead of working directly with coordinates, the formula uses vector expressions, which makes it cleaner and more flexible.
If a point divides a segment between points A and B, the section formula tells exactly where that point lies along the line.
2. Internal Division Formula
Suppose a point P divides the line segment joining A and B in the ratio \(m:n\) internally.
If the position vectors of A and B are:
\[ \vec{a}, \quad \vec{b} \]
then the position vector of P is:
\[ \vec{p} = \dfrac{n\vec{a} + m\vec{b}}{m + n} \]
This formula gives a point lying between A and B.
2.1. Example
Let:
\[ \vec{a} = \langle 2, 3 \rangle, \quad \vec{b} = \langle 8, 7 \rangle \]
If P divides AB in the ratio \(2:1\), then:
\[ \vec{p} = \dfrac{1\vec{a} + 2\vec{b}}{2 + 1} = \dfrac{\langle 2, 3 \rangle + \langle 16, 14 \rangle}{3} = \langle 6, 17/3 \rangle \]
3. External Division Formula
If P divides AB externally in the ratio \(m:n\), then P lies outside the segment AB. The vector formula becomes:
\[ \vec{p} = \dfrac{n\vec{a} - m\vec{b}}{n - m} \]
External division creates a point that is aligned with A and B but beyond the segment.
3.1. Example
Let:
\[ \vec{a} = \langle 1, 2 \rangle, \quad \vec{b} = \langle 5, 6 \rangle \]
If P divides AB externally in the ratio \(3:1\), then:
\[ \vec{p} = \dfrac{1\vec{a} - 3\vec{b}}{1 - 3} = \dfrac{\langle 1, 2 \rangle - \langle 15, 18 \rangle}{-2} \]
Simplifying:
\[ \vec{p} = \dfrac{\langle -14, -16 \rangle}{-2} = \langle 7, 8 \rangle \]
4. Midpoint Formula (Special Case)
The midpoint is a special case of internal division where the ratio is \(1:1\). If M is the midpoint of AB, then:
\[ \vec{m} = \dfrac{\vec{a} + \vec{b}}{2} \]
4.1. Example
If:
\[ \vec{a} = \langle 4, 1 \rangle, \quad \vec{b} = \langle 10, 7 \rangle \]
Then midpoint M has:
\[ \vec{m} = \dfrac{\langle 4, 1 \rangle + \langle 10, 7 \rangle}{2} = \langle 7, 4 \rangle \]
5. Why the Section Formula Works
The section formula is essentially a weighted average of the vectors. A point closer to A gets more weight from \(\vec{a}\), and a point closer to B gets more weight from \(\vec{b}\).
This ‘weighted balance’ interpretation makes the formula intuitive and useful in geometry, vectors, and coordinate calculations.
6. More Examples
- Internal division of \(\vec{a}\) and \(\vec{b}\): weighted average.
- External division: weighted difference.
- Midpoint: both vectors equally weighted.