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Standard XII

Mathematics

Section Formula

Show that the...

Question

Show that the positive vector of the point $P$ , which divides the line joining the points $A$ and $B$ having position vectors $\to a$ and $\to b$ internally in ratio $m : n$ is $\frac{m \to b + n \to a}{m + n}$ .

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Solution

Let $O$ be the origin representing the position vectors $O A$ and $O B$ with respect to the two points $A$ and $B$ . Let $A$ and $B$ be divided by a third point $P$ internally in the ratio $m : n$
$∴ \frac{A P}{A B} = \frac{m}{n}$ or $n A P = m P B$
$\Rightarrow n \to A P = m \to P B$
$\Rightarrow n (\to O P - \to O A) = m (\to O B - \to O P)$
$\Rightarrow n (\to r - \to a) = m (\to b - \to r)$
$n \to r - n \to a = m \to b - m \to r$
$∴ m \to r + n \to r = m \to b + n \to a$
or $\to r = \frac{m \to b + n \to a}{m + n}$ (proved).

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