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Vector Algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector Algebra2 Marks
Question
Find the position vector (externally) 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.
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Answer

Given that $\vec{{P}}=\hat{1}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\vec{\mathrm{Q}}=-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ 
The $\vec{\mathrm{R}}$ does not lie on the segment PQ (external division). If $\mathrm{m}: n$ is the ratio in which $\vec{\mathrm{R}}$ divides PQ, then
$\vec{\mathrm{R}}=\frac{m \vec{\mathrm{Q}}-n \vec{\mathrm{P}}}{m-n}$ 
Given m : n = 2 : 1, m = 2 and n = 1
$\Rightarrow \vec{\mathrm{R}}=\frac{2(-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})-1(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})}{2-1}$ = $\frac{-3 \hat{1}+0 \hat{\jmath}+3 \hat{k}}{1}=-3 \hat{1}+3 \hat{k}$

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