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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 \[\overrightarrow P = \dfrac{{m\overrightarrow b + n\overrightarrow a }}{{n + m}}\]

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Last updated date: 11th Jun 2024
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Answer
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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.

Complete step-by-step answer:
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. \[\overrightarrow {AB} = - \overrightarrow {BA} \,\]