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Vector Algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector Algebra2 Marks
Question
Find the position vector (internally) 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 $\vec{\mathrm{R}}$ lies on the segment PQ(internal division). If $\mathrm{m}: \mathrm{n}$ is the ratio in which $\vec{\mathrm{R}}$ divides $\mathrm{PQ}$ then
$\vec{\mathrm{R}}=\frac{\mathrm{m} \vec{\mathrm{Q}}+\mathrm{n} \vec{\mathrm{P}}}{\mathrm{m}+\mathrm{n}}$ 
Given m : n = 2 : 1, m = 2 and n = 1
$\Rightarrow \vec{\mathrm{R}}=\frac{2(-\hat{\imath}+\hat{\jmath}+\hat{\mathrm{k}})+1(\hat{\imath}+2 \hat{\jmath}-\hat{\mathrm{k}})}{2+1}$ = $\frac{-1 \hat{1}+4 \hat{\jmath}+\hat{k}}{3}=\frac{-1}{3} \hat{i}+\frac{4}{3} \hat{\jmath}+\frac{1}{3} \hat{k}$ 

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