Question

L and M are the two points with position vectors $2\overrightarrow a - \overrightarrow b$ and $\overrightarrow a + 2\overrightarrow b$ respectively. Write the position vector of a point N which divides the line segment LM in the ratio 2:1 externally.

Answer 1

Consider 'R' be the position vector of a point N which divides the line LM in the ratio 2:1 externally which can be found out by formula $\overrightarrow R = \dfrac{{m\overrightarrow M - n\overrightarrow L }}{{m - n}}$.

Complete step-by-step solution -
Given data:
L and M are the two points with position vectors $2\vec a - \vec b$ and $\vec a + 2\vec b$ respectively.
So, $\vec L = 2\vec a - \vec b$ and $\vec M = \vec a + 2\vec b$
Now consider a position vector (say $\vec R$) of point N which divides the line segment LM in the ratio 2:1 externally.
Now as we know that this position vector $\vec R$ is calculated as, $\vec R = \dfrac{{m\vec M - n\vec L}}{{m - n}}$.................... (1), where m and n are the values of the ratio which divide the line externally, i.e. m: n = 2: 1.
Therefore, m = 2, and n = 1.
Now substitute the values we have,
$\Rightarrow \vec R = \dfrac{{2\left( {\vec a + 2\vec b} \right) - 1\left( {2\vec a - \vec b} \right)}}{{2 - 1}}$
Now simplify this we have,
$\Rightarrow \vec R = \dfrac{{\left( {2\vec a + 4\vec b} \right) - 1\left( {2\vec a - \vec b} \right)}}{1}$
$\Rightarrow \vec R = 2\vec a + 4\vec b - 2\vec a + \vec b$
$\Rightarrow \vec R = 5\vec b$
So this is the required position vector of a point N which divides the line segment LM in the ratio 2:1 externally. So this is the required answer.

Note: Whenever we face such types of questions the key concept we have to remember is that always recall the position vector formula if a point N divides the line segment LM in the ratio m:n externally which is stated above so simply substitute the values in this formula and simplify we will get the required answer.