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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA3 Marks
Question
$\text{If}\ \vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\ \vec{\text{b}}=2\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$ and $\vec{\text{c}}=\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}},$ find a unit vector parallel to the vector $2\vec{\text{a}}-\vec{\text{b}}+3\vec{\text{c}}.$
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Answer

$\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\ \vec{\text{b}}=2\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$ and $\vec{\text{c}}=\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}$

$2\vec{\text{a}}-\vec{\text{b}}+3\vec{\text{c}}=2\big(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}\big)-\big(2\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}\big)+3\big(\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}\big)$

$=2\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}-2\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}}+3\hat{\text{i}}-6\hat{\text{j}}+3\hat{\text{k}}$

$=3\hat{\text{i}}-3\hat{\text{j}}+2\hat{\text{k}}$

$\Big|2\vec{\text{a}}-\vec{\text{b}}+3\vec{\text{c}}\Big|=\sqrt{3^{2}+(-3)^{2}+2^{2}}$ $=\sqrt{9+9+4}=\sqrt{22}$

Hence, the unit vector along $2\vec{\text{a}}-\vec{\text{b}}+3\vec{\text{c}}\ \text{is}$

$\frac{2\vec{\text{a}}-\vec{\text{b}}+3\vec{\text{c}}}{\Big|2\vec{\text{a}}-\vec{\text{b}}+3\vec{\text{c}}\Big|}=\frac{3\hat{\text{i}}-3\hat{\text{j}}+2\hat{\text{k}}}{\sqrt{22}}$ $\frac{3}{\sqrt{22}}\hat{\text{i}}-\frac{3}{\sqrt{22}}\hat{\text{j}}+\frac{2}{\sqrt{22}}\hat{\text{k}}.$

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