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

Last updated date: 11th Jun 2024
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Hint: Position Vector: Position vector is nothing but a straight line whose one end is fixed to a body and the other end is attached to a morning point which is used to describe the position of that body relative to the body.
Triangle law of vector addition: In a triangle, two directions are taken in order while the third one is in the opposite direction. Therefore the sum of 2 sides taken in order is equal to the third side which is taken in the opposite direction.

Let 2 points A & B and P is the point on the line AB and O is the origin then position vector of OA is $\overrightarrow a$ and OB is $\overrightarrow b$ and m:n is the ratio in which P divides A & B and OP will be $\overrightarrow P$.
Here $\dfrac{{AP}}{{PB}} = \dfrac{m}{n} - - - - - - - - - - - (i)$
$\Rightarrow$$n.AP = m.PB$
In vector notation, $n. \overrightarrow {AP} = m.\overrightarrow {PB} - - - - - - - - - - (II)$
Using triangle law of vector addition in OPA, we get $\overrightarrow {OP} = \overrightarrow {OA} + \overrightarrow {AP}$
$\Rightarrow$$\overrightarrow {AP} = \overrightarrow {OP} - \overrightarrow {OA} - - - - - - - - - - - (III)$
And in OPB= $\overrightarrow {OB} = \overrightarrow {OP} + \overrightarrow {PB}$
$\Rightarrow$$\overrightarrow {PB} = \overrightarrow {OB} - \overrightarrow {OP} - - - - - - - - - - - (IV)$
Put the value of $\overrightarrow {AP}$ and $\overrightarrow {PB}$ from $(III)$& $(IV)$ in $(II)$
$\Rightarrow$$n\left( {\overrightarrow {OP} - \overrightarrow {OA} } \right) = m\left( {\overrightarrow {OB} - \overrightarrow {OP} } \right)$
$\Rightarrow$$n\left( {\overrightarrow P - \overrightarrow a } \right) = m\left( {\overrightarrow b - \overrightarrow P } \right)$
$\Rightarrow$$n\overrightarrow P - n\overrightarrow a = m\overrightarrow b - m\overrightarrow P$
$\Rightarrow$$\overrightarrow P (n + m) = m\overrightarrow b + n\overrightarrow a$
$\Rightarrow$$\overrightarrow P = \dfrac{{m\overrightarrow b + n\overrightarrow a }}{{n + m}}$

Note: 1) Position vector can be written as the sum of 2 vectors
E.g.$\overrightarrow {AB} = \overrightarrow {AP} - \overrightarrow {PB}$
2) Value of position vector is negative if we opposite the direction
e.g. $\overrightarrow {AB} = - \overrightarrow {BA} \,$