NIMCET 2020 — Mathematics PYQ

If a point P divides a line joining points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) in a ratio of m:n, then
xₚ = $\frac{n{x}_1+m{x}_2}{n+m}$, yₚ = $\frac{n{y}_1+m{y}_2}{n+m}$ and zₚ = $\frac{n{z}_1+m{z}_2}{n+m}$

Calculation:

Assuming,
$\vec{a}$ = xₐi + yₐj + zₐk
$\vec{b}$ = x₂i + y₂j + z₂k

Vector of point p is
$\vec{p}$ = xₚi + yₚj + zₚk

∵ xₚ = $\frac{n{x}_1+m{x}_2}{n+m}$, yₚ = $\frac{n{y}_1+m{y}_2}{n+m}$ and zₚ = $\frac{n{z}_1+m{z}_2}{n+m}$
∴ $\vec{p}$ = $\frac{n{x}_1+m{x}_2}{n+m}$ i + $\frac{n{y}_1+m{y}_2}{n+m}$ j + $\frac{n{z}_1+m{z}_2}{n+m}$ k

\begin{aligned}
 & \Rightarrow\mathrm{\vec{p}=\frac{n\left(x_{a}i+y_{a}j+z_{a}k\right)+m\left(x_{b}i+y_{b}j+z_{b}k\right)}{n+m}} \\
 & \Rightarrow\vec{\mathbf{p}}=\frac{\mathrm{n\vec{a}+m\vec{b}}}{\mathrm{n+m}}
\end{aligned}