Question
If point $C (\bar{c})$ divides the segment joining the points $A (\bar{a})$ and $B (\bar{b})$ internally in the ratio $m: n$, then prove that
$\bar{c}=\frac{m \bar{b}+n \bar{a}}{m+n}$
$\bar{c}=\frac{m \bar{b}+n \bar{a}}{m+n}$
✓
Answer
Given that $C (\vec{c})$ divides the segment joining the points $A (\vec{a})$ and $B (\vec{b})$ internally in the ratio $m: n$
We need to prove that $\vec{c}=\frac{m \vec{b}+n \vec{a}}{m+n}$
Consider the following figure
since
$n \overrightarrow{A C}=m \overrightarrow{C B}$
$n(\overrightarrow{O C}-\overrightarrow{O A})=m(\overrightarrow{O B}-\overrightarrow{O C})$
$n(\vec{c}-\vec{a})=m(\vec{b}-\vec{c})$
$n \vec{c}-n \vec{a}=m \vec{b}-m \vec{c}$
$(n+m) \vec{c}=m \vec{b}+n \vec{a}$
$\vec{c}=\frac{m \vec{b}+n \vec{a}}{m+n}$
Hence proved.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find the value of k for which the function $\text{f(x)}=\begin{cases}\frac{\text{x}^{2} + 3\text{x} - 10}{\text{x} - 2},&\text{x}\neq2\\\text{k},&\text{x} = {2}\end{cases}$ is continues at x = 2.
→Solve the following equations by reduction method $: 2x + y = 5, 3x + 5y = -3$
→If $y = x +\tan x$, show that $\cos ^2 x \cdot \frac{d^2 y}{d x^2}-2 y+2 x=0$
→Form the differential equation of:all circles which pass through the origin and whose centres lie on X-axis.
Find the equation of rthe planes parallel to the plane x + 2y - 2z + 8 = 0 that are at a distance of 2 units from the point (2, 1, 1).
Show that $\text{f}(\text{x})=\text{e}^{\frac{1}{\text{x}}},\text{x}\neq0$ is a decreasing function for all $\text{x}\neq0.$
→Show that $\tan ^{-1} \frac{1}{2}=\frac{1}{3} \tan ^{-1} \frac{11}{2}$.
→If $\text{y}=\log\Big(\sqrt{\text{x}}+\frac{1}{\sqrt{\text{x}}}\Big),$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\text{x}-1}{2\text{x}(\text{x}+1)}$
→Differentiate $\sin ^{-1}\left(\frac{2 \cos x+3 \sin x}{\sqrt{13}}\right)$ w.r. to $x$
→Find the vector equations of the following planes in scalar product form $(\vec{\text{r}}\cdot\vec{\text{n}}=\text{d}):$
$\vec{\text{r}}=\hat{\text{i}}-\hat{\text{j}}+\lambda(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})+\mu(4\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}})$
→$\vec{\text{r}}=\hat{\text{i}}-\hat{\text{j}}+\lambda(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})+\mu(4\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}})$