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Home  >>  CBSE XII  >>  Math  >>  Vector Algebra
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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 \( \hat i + 2\hat j − \hat k\) and \( – \hat i + \hat j + \hat k\) respectively, in the ratio $2 : 1$ internally

This question has multiple parts. Therefore each part has been answered as a separate question on Clay6.com

$\begin{array}{1 1}(A) \large\frac{\hat i + 4\hat j - \hat k}{3} \\(B) \large\frac{-\hat i + 4\hat j + \hat k}{2} \\ (C) \large\frac{-\hat i + 4\hat j + \hat k}{3} \\ (D) \large\frac{\hat i - 4\hat j - \hat k}{3} \end{array} $

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  • Section formula:When a point R divides a line segment in the ratio m:n internally,then $\overrightarrow r=\large\frac{m\overrightarrow b+n\overrightarrow a}{m+n}$
Step 1:
Let $\overrightarrow a=\hat i+2\hat j-\hat k$ and $\overrightarrow b=-\hat i+\hat j+\hat k$.
$\overrightarrow {OR}$ divides $\overrightarrow {PQ}$ in the ratio 2:1.
Let us discuss when $\overrightarrow{OR}$ divides $\overrightarrow{PQ}$ internally.
We know when $\overrightarrow{OR}$ divides internally in the ratio m:n,then $\overrightarrow {OR}=\large\frac{m\overrightarrow b+n\overrightarrow a}{m+n}$
Step 2:
Here $\overrightarrow a=\hat i+2\hat j-\hat k$ and $\overrightarrow b=-\hat i+\hat j+\hat k$ and m=2 and n=1.
$\overrightarrow {OR}=\large\frac{2(-\hat i+\hat j+\hat k)+1(\hat i+2\hat j-\hat k)}{2+1}$
$\qquad=\large\frac{-2\hat i+2\hat j+2\hat k+\hat i+2\hat j-\hat k}{3}$
$\qquad=\large\frac{-\hat i + 4\hat j + \hat k}{3}$
answered May 20, 2013 by sreemathi.v
edited May 20, 2013 by sreemathi.v
 

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