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

Hint: 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
\[\begin{align}
  & \Rightarrow \bar{a}=\dfrac{2\left( x\bar{a}+y\bar{b} \right)+3\left( \bar{a}+2\bar{b} \right)}{2+3} \\
 & \Rightarrow \bar{a}=\dfrac{2x\bar{a}+2y\bar{b}+3\bar{a}+6\bar{b}}{5} \\
\end{align}\]
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
\[\begin{align}
  & \Rightarrow \dfrac{2y+6}{5}=0 \\
 & \Rightarrow 2y=-6 \\
 & \Rightarrow y=-3 \\
\end{align}\]
Now, by comparing the co – efficient of \[\bar{a}\] we will get
\[\begin{align}
  & \Rightarrow \dfrac{2x+3}{5}=1 \\
 & \Rightarrow 2x+3=5 \\
 & \Rightarrow 2x=2 \\
 & \Rightarrow x=1 \\
\end{align}\]
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.