You visited us 11 times! Enjoying our articles? Unlock Full Access!

Position Vector

If point Cc...

Question

If point $C (¯ ¯ c)$ divides the segment joining the point $A (¯ ¯ ¯ a)$ and $B (¯ ¯ b)$ internally in the ratio $m : n$ , then prove that $¯ ¯ c = \frac{m ¯ ¯ b + n ¯ ¯ ¯ a}{m + n}$

Open in App

Solution

Let $O$ be the origin.
Then $- - \to A O = \to a$ and $- - \to O B = \to b$
Let $P$ be a point on $- - \to A B$ such that $\frac{- - \to A P}{- - \to P B} = \frac{m}{n}$
$\Rightarrow n (- - \to A P) = m (- - \to P B)$
$\Rightarrow n (- - \to O P - - - \to O A) = m (- - \to O B - - - \to O P)$
$\Rightarrow (m + n) O P = m O B + n O A$
$\Rightarrow (m + n) - - \to O P = m \to b + n \to a$
$\Rightarrow - - \to O P = \to c = \frac{m \to b + n \to a}{m + n}$

Suggest Corrections

Similar questions

P

A

B

\to a

\to b

m : n

\frac{m \to b + n \to a}{m + n}

(6, 4)

(1, - 7)

m : n

m + n

P

A

B

\to a

\to b

m : n

\frac{m \to b + n \to a}{m + n}

Which of the follwing statements are correct ?
1. The coordinates of the point R which divides the line

segment joining two points $P (x_{1}, y_{1}, z_{1})$ and $Q (x_{2}, y_{2}, z_{2})$

externally in the ratio m:n are $(\frac{m x_{2} + n x_{1}}{m + n}, \frac{m y_{2} + n y_{1}}{m + n}, \frac{m z_{2} + n z_{1}}{m + n})$ .

2. If R divides PQ internally in the ratio m:n, then its coordinates

are obtained by replacing n by -n in the statement 1.

P (ˆ i + 2 ˆ j - ˆ k)

Q (- ˆ i + ˆ j + ˆ k)

\frac{- 1}{3} ˆ i + \frac{4}{3} ˆ j + \frac{1}{3} ˆ k

\to r = \frac{m \to b - n \to a}{m - n}

Join BYJU'S Learning Program