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Standard XII

Physics

Position Vector

Show that the...

Question

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

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Solution

$\to r$ divides $\to a, \to b$ in ratio $λ : 1$
$\to P B$ is parallel to $\to A P$
$\vec { AP } =\lambda \vec { PB }
$\to O P - \to O A = λ (\to O B - \to O P)$
$\to r - \to a = λ (\to b - \to r)$
$\to r (λ + 1) = λ \to b + \to a$
$\to r = \frac{m \to b + n \to a}{m + n}$

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Q. Show that the positive vector of the point

P

, which divides the line joining the points

A

and

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m : n

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

Q. Find the position vector of a vector of a point R which divides the line joining two points P and Q whose position vectors are

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Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $^i + 2^j -^k$ and $-^i +^j +^k$ respectively, in the ratio 2:1
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in the ratio 2 : 1 internally is

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Reason:

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

Q. Find the position vector of a point R which divides the line joining the two points P and Q with position vectors

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Show that the position vector of the point P, which divides the line joining the points A and B having position vector →a and →b internally in the ratio m:n is m→b+n→am+n.

→r divides →a,→b in ratio λ:1→PB is parallel to →AP$\vec { AP } =\lambda \vec { PB } →OP−→OA=λ(→OB−→OP)→r−→a=λ(→b−→r)→r(λ+1)=λ→b+→a→r=m→b+n→am+n

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

$\to r$ divides $\to a, \to b$ in ratio $λ : 1$
$\to P B$ is parallel to $\to A P$
$\vec { AP } =\lambda \vec { PB }
$\to O P - \to O A = λ (\to O B - \to O P)$
$\to r - \to a = λ (\to b - \to r)$
$\to r (λ + 1) = λ \to b + \to a$
$\to r = \frac{m \to b + n \to a}{m + n}$