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

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Solution

$\to a =^i + 2^j -^k \to b = -^i +^j +^k$
Position vector of a point for internal division $\frac{m \to b + n \to a}{m + n} = \frac{2 (-^i +^j +^k) + 1 (^i + 2^j -^k)}{2 + 1} = \frac{-^i + 4^j +^k}{3}$
Position vector of a point for external division
$\frac{m \to b + n \to a}{m - n} = \frac{- 3^i + 3^k}{2 - 1} = - 3^i + 3^k$

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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
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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

→a=^i+2^j−^k→b=−^i+^j+^kPosition vector of a point for internal division m→b+n→am+n=2(−^i+^j+^k)+1(^i+2^j−^k)2+1=−^i+4^j+^k3Position vector of a point for external divisionm→b+n→am−n=−3^i+3^k2−1=−3^i+3^k

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 $2 : 1$
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