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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA3 Marks
Question
Write the projection of $\vec{\text{b}}+\vec{\text{c}}$ on $\vec{\text{a}}$ when $\vec{\text{a}}=2\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-2\hat{\text{k}}$ and $\vec{\text{c}}=2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}.$
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Answer

Given that
$\vec{\text{a}}=2\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}$
$\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-2\hat{\text{k}}$
$\vec{\text{c}}=2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}$
$\vec{\text{b}}+\vec{\text{c}}=\hat{\text{i}}+2\hat{\text{j}}-2\hat{\text{k}}+2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}$
$=3\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}$
Projection of $\vec{\text{b}}+\vec{\text{c}}$ on $\vec{\text{a}}$ is
$\frac{\big(\vec{\text{b}}+\vec{\text{c}}\big).\vec{\text{a}}}{|\vec{\text{a}}|}$
$=\frac{\big(3\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}\big).\big(2\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}\big)}{2\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}}$
$=\frac{6-2+2}{\sqrt{4+4+1}}$
$=\frac{6}{3}$
$=2$

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