Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are i + 2 j - k and - i + j + k respectively, in the ratio 2 : 1 (i) internally (ii) externally

Here, a = i + 2 j - k and b = - i + j + k (i) The position vector of R dividing the join of P and Q internally in the ratio 2 : 1 is R = m + n = 2 + 1 = 2 + 1 = - 3 i + 3 j + 3 k (ii) The position vector of R dividing the join of P and Q externally in the ratio 2 : 1 is R = m - n = 2 - 1 = 2 - 1 = - 3 i + 0 j + 3 k = - 3 i + 3 k

Vector Algebra — Class 12 Maths Solution

ncert exercise SA NCERT,Page 440,Ex.10.2,Q.No 15

Question

Find the position vector of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $\hat i + 2\hat j - \hat k$ and $- \hat i + \hat j + \hat k$ respectively, in the ratio 2 : 1

(i) internally

(ii) externally

Step-by-step Solution

Here, $\vec a = \hat i + 2\hat j - \hat k$ and $\vec b = - \hat i + \hat j + \hat k$

(i) The position vector of $R$ dividing the join of $P$ and $Q$ internally in the ratio 2 :

1 is
$\vec R = \cfrac{{m\vec b + n\vec a}}{{m + n}} = \cfrac{{2(\vec b) + 1(\vec a)}}{{2 + 1}}$

$= \cfrac{{2( - \hat i + \hat j + \hat k) + 1(\hat i + 2\hat j - \hat k)}}{{2 + 1}} = - \cfrac{1}{3}\hat i + \cfrac{4}{3}\hat j + \cfrac{1}{3}\hat k$

(ii) The position vector of $R$ dividing the join of $P$ and $Q$ externally in the ratio 2 :

1 is
$\vec R = \cfrac{{m\vec b - n\vec a}}{{m - n}} = \cfrac{{2(\vec b) - 1(\vec a)}}{{2 - 1}} = \cfrac{{2( - \hat i + \hat j + \hat k) - 1(\hat i + 2\hat j - \hat k)}}{{2 - 1}}$
$= - 3\hat i + 0\hat j + 3\hat k = - 3\hat i + 3\hat k$

← All Vector Algebra solutions Practice tests →

NCERT & Exemplar solution for CBSE Class 12 Mathematics, Vector Algebra. Curated by Sachin Sharma. Free for all students.