Question

The position vectors of P and Q are respectively a and b. If R is a point on PQ, such that PR=5PQ, then the position vector of R is
A) 5b - 4a
B) 5b + 4a
C) 4b - 5a
D) 4b + 5a

Answer 1

Hint: Position vectors of P and Q are given, we can let the position vector of R as any variable vector. It lies on PQ according to a certain condition, we can put this variable into the condition to calculate its value and thus the required answer will be obtained.

Complete step-by-step answer:

The position of vectors P and Q are given to be a and b respectively.
 $
  \Rightarrow P = \vec a \\
  \Rightarrow Q = \vec b \;
  $
R is a point on PQ, let its position vector be c
 $ \Rightarrow R = \vec c $
This point R lies on PQ such that PR = 5PQ (given). In vectors, it can be written as:
 $ \overrightarrow {PR} = 5\overrightarrow {PQ} ....(1) $
Now,
 $ \overrightarrow {PR} $ = Position vector of R – Position vector of P
 $ \overrightarrow {PQ} $ = Position vector of Q – Position vector of P
 $
   \Rightarrow \overrightarrow {PR} = R - P \\
   \Rightarrow \overrightarrow {PQ} = Q - P \;
  $
Substituting the values, we get:
 $
  \Rightarrow \overrightarrow {PR} = \vec c - \vec a \\
  \Rightarrow \overrightarrow {PQ} = \vec b - \vec a \\
  $
Form (1), we have:
 $
  \overrightarrow {PR} = 5\overrightarrow {PQ} \\
   \Rightarrow \vec c - \vec a = 5\left( {\vec b - \vec a} \right) \\
  \Rightarrow \vec c - \vec a = 5\vec b - 5\vec a \\
  \Rightarrow \vec c = 5\vec b - 5\vec a + \vec a \\
  \Rightarrow \vec c = 5\vec b - 4\vec a \;
  $
Position vector of R is c and its value is 5b – 4a (without vector sign)
Therefore, the position vector of R is $5b – 4a$ and the correct option is A).
So, the correct answer is “Option A”.

Note: Vectors are the quantities that have both magnitude and direction and are represented by an over right arrow on head. The position vector specifies the position of a vector or a point. So, they can be represented with or without vector signs. It is always better to represent them with the vector sign but as given in the options, it can also be skipped.
This question can also be solved by substituting the given values at last like:
Given: $ PR = 5PQ $
For finding the value of R, this can be written as:
 $
  R - P = 5(Q - P) \\
  R = 5Q - 5P + P \\
  R = 5Q - 4P \;
  $
The position vectors of P and Q are a and b respectively, substituting this, we get:
 $ R = 5b - 4a $