Question

If the position vector of the point P is $\bar{a}+2\bar{b}$ and A ($\bar{a}$) divides PQ internally in the ratio 2:3, then the position vector of Q is
(a) $\bar{a}+\bar{b}$
(b) $2\bar{a}-\bar{b}$
(c) $\bar{a}-3\bar{b}$
(d) $\bar{b}-2\bar{a}$

Answer 1

Here we simply use the internal division formula and evaluate the required result. Let us consider that the position vector of Q as $x\bar{a}+y\bar{b}$ and if A divides PQ in the ratio m:n then we write the internal division formula as
$\bar{A}=\dfrac{m\bar{Q}+n\bar{P}}{m+n}$ and them=n by comparing the coefficient of $\bar{a}$ and $\bar{b}$ we get the position vector of Q.

Complete step-by-step solution
Let us consider the given position vector of P as
$\bar{P}=\bar{a}+2\bar{b}$
Let us consider that the position vector of A as
$\bar{A}=\bar{a}$
Let us assume that the position vector of Q as
$\bar{Q}=x\bar{a}+y\bar{b}$
Now we know that the internal division formula if A divides PQ in the ratio m:n as
$\bar{A}=\dfrac{m\bar{Q}+n\bar{P}}{m+n}$
Now, by substituting the values of P, Q, A and taking the ratio as 2:3 the values of m, n are 2, 3 respectively we will get

& \Rightarrow \bar{a}=\dfrac{2\left( x\bar{a}+y\bar{b} \right)+3\left( \bar{a}+2\bar{b} \right)}{2+3} \\\ & \Rightarrow \bar{a}=\dfrac{2x\bar{a}+2y\bar{b}+3\bar{a}+6\bar{b}}{5} \\\ \end{aligned}$$ By writing the terms of $$\bar{a}$$ and $$\bar{b}$$ separately we will get $$\Rightarrow \bar{a}=\bar{a}\left( \dfrac{2x+3}{5} \right)+\bar{b}\left( \dfrac{2y+6}{5} \right)$$ By comparing the co – efficient of $$\bar{b}$$ we will get $$\begin{aligned} & \Rightarrow \dfrac{2y+6}{5}=0 \\\ & \Rightarrow 2y=-6 \\\ & \Rightarrow y=-3 \\\ \end{aligned}$$ Now, by comparing the co – efficient of $$\bar{a}$$ we will get $$\begin{aligned} & \Rightarrow \dfrac{2x+3}{5}=1 \\\ & \Rightarrow 2x+3=5 \\\ & \Rightarrow 2x=2 \\\ & \Rightarrow x=1 \\\ \end{aligned}$$ By substituting the values of $$x$$ and $$y$$ we will get the position vector of Q as $$\bar{Q}=\bar{a}-3\bar{b}$$ **Therefore option (c) is the correct answer.** **Note:** Many students will do mistake in applying the division formula by taking in the sequence that is instead of writing the formula as $$\bar{A}=\dfrac{m\bar{Q}+n\bar{P}}{m+n}$$, due to confusion they will write as$$\bar{A}=\dfrac{m\bar{P}+n\bar{Q}}{m+n}$$. So students need to take care of applying the formula. This is the only point where students can make mistakes.