Section formula
\[\vec r=\frac{m\vec b+n\vec a}{m+n}\]
Vector Algebra
Components and Section Formula
Understand
The vector joining \(P(x_1,y_1,z_1)\) to \(Q(x_2,y_2,z_2)\) is \(\overrightarrow{PQ}=(x_2-x_1)\hat i+(y_2-y_1)\hat j+(z_2-z_1)\hat k\).
Section formula
If \(R\) divides \(PQ\) in the ratio \(m:n\) (position vectors \(\vec a,\vec b\)):
\[\text{internally: }\vec r=\frac{m\vec b+n\vec a}{m+n},\qquad \text{externally: }\vec r=\frac{m\vec b-n\vec a}{m-n}\]
The midpoint has position vector \(\frac{\vec a+\vec b}{2}\), and the centroid of a triangle \(\frac{\vec a+\vec b+\vec c}{3}\).
Key Concepts
- PQ = (position vector of Q) − (position vector of P).
- Section formula: weights cross over (m goes with b).
- Midpoint = average.
Formula Bank
Common Mistakes
Section formula weights
For ratio \(m:n\) from \(A\) to \(B\), the weight \(m\) goes with \(\vec b\) (the far point).
Practice & Topic Test
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Use differences of position vectors for displacements; weighted averages for division points.
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