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Algebra of Vectors — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Vectors2 Marks
Question
Write a unit vector in the direction of the sum of the vectors $\vec{\text{a}}=2\hat{\text{i}}+2\hat{\text{j}}-5\hat{\text{k}}$ and $\vec{\text{b}}=2\hat{\text{i}}+\text{y}\hat{\text{j}}-7\hat{\text{k}}$.
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Answer

We have, $\vec{\text{a}}=2\hat{\text{i}}+2\hat{\text{j}}-5\hat{\text{k}}$ and $\vec{\text{b}}=2\hat{\text{i}}+\text{y}\hat{\text{j}}-7\hat{\text{k}}$
$\therefore\ \vec{\text{a}}+\vec{\text{b}}=\big(2\hat{\text{i}}+2\hat{\text{j}}-5\hat{\text{k}}\big)+\big(2\hat{\text{i}}+\hat{\text{j}}-7\hat{\text{k}}\big)$
$=4\hat{\text{i}}+3\hat{\text{j}}-12\hat{\text{k}}$
$\Rightarrow\ \big|\vec{\text{a}}+\vec{\text{b}}\big|=\sqrt{4^2+3^2+(-12)^2}$
$=\sqrt{16+9+144}$
$=\sqrt{169}$
$=13$
$\therefore$ Required unit vector $=\frac{\vec{\text{a}}+\vec{\text{b}}}{\big|\vec{\text{a}}+\vec{\text{b}}\big|}=\frac{4\hat{\text{i}}+3\hat{\text{j}}-12\hat{\text{k}}}{13}$
$=\frac{4}{13}\hat{\text{i}}+\frac{3}{13}\hat{\text{j}}-\frac{12}{13}\hat{\text{k}}$

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