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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 $2 : 1$ externally

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

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

1 Answer

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

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