Question

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 $$\overrightarrow{P}={\displaystyle \frac{m\overrightarrow{b}+n\overrightarrow{a}}{n+m}}$$

Answer 1

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 $${\displaystyle \frac{AP}{PB}}={\displaystyle \frac{m}{n}}-----------(i)$$
$$\Rightarrow$$$$n.AP=m.PB$$
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 $$(II)$$
$$\Rightarrow$$$$n\left(\overrightarrow{OP}-\overrightarrow{OA}\right)=m\left(\overrightarrow{OB}-\overrightarrow{OP}\right)$$
$$\Rightarrow$$$$n\left(\overrightarrow{P}-\overrightarrow{a}\right)=m\left(\overrightarrow{b}-\overrightarrow{P}\right)$$
$$\Rightarrow$$$$n\overrightarrow{P}-n\overrightarrow{a}=m\overrightarrow{b}-m\overrightarrow{P}$$
$$\Rightarrow$$$$\overrightarrow{P}(n+m)=m\overrightarrow{b}+n\overrightarrow{a}$$
$$\Rightarrow$$$$\overrightarrow{P}={\displaystyle \frac{m\overrightarrow{b}+n\overrightarrow{a}}{n+m}}$$

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. $$\overrightarrow{AB}=-\overrightarrow{BA}$$