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VECTOR ALGEBRA — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSVECTOR ALGEBRA2 Marks
Question
For given vectors, $\vec{a}=2\hat{i}-\hat{j}+2\hat{k}\ \ \text{and}\ \vec{b}=-\hat{i}+\hat{j}-\hat{k},$ find the unit vector in the direction of the vector $\vec{a}+\vec{b}.$
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Answer

The given vectors are $\vec{a}=2\hat{i}-\hat{j}+2\hat{k}\ \ \text{and}\ \vec{b}=-\hat{i}+\hat{j}-\hat{k}$ $\vec{a}=2\hat{i}-\hat{j}+2\hat{k}$ $\vec{b}=-\hat{i}+\hat{j}-\hat{k}$ $\therefore\vec{a}+\vec{b}=(2-1)\hat{i}+(-1+1)\hat{j}+(2-1)\hat{k}$ $=1\hat{i}+0\hat{j}+1\hat{k}=\hat{i}+\hat{k}$ $\big|\vec{a}+\vec{b}\big|=\sqrt{1^2+1^2}=\sqrt{2}$Hence, the unit vector in the direction of $\big(\vec{a}+\vec{b}\big)$ is
$\frac{\big(\vec{a}+\vec{b}\big)}{\big|\vec{a}+\vec{b}\big|}=\frac{\hat{i}+\hat{k}}{\sqrt{2}}=\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{k}$

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