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Algebra of Vectors — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Vectors2 Marks
Question
If $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}},\ \vec{\text{b}}=\hat{\text{j}}+\hat{\text{k}},\ \vec{\text{c}}=\hat{\text{k}}+\hat{\text{i}}$, find the unit vector in the direction of $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}$.
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Answer

Let $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}},\ \vec{\text{b}}=\hat{\text{j}}+\hat{\text{k}},\ \vec{\text{c}}=\hat{\text{k}}+\hat{\text{i}}$
Then, $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{j}}+\hat{\text{k}}+\hat{\text{k}}+\hat{\text{i}}$
$=2\big(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}\big)$
$\therefore\ |\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}|=\sqrt{2^2+2^2+2^2}$
$=\sqrt{12}$
$=2\sqrt3$
Therefore, unit vector in the direction of $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\frac{2\big(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}\big)}{2\sqrt3}=\frac{1}{\sqrt3}\big(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}\big)$

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