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Scalar Or Dot Product — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsScalar Or Dot Product3 Marks
Question
Find the projection of $\vec{\text{b}}+\vec{\text{c}}$ on $\vec{\text{a}},$ where $\vec{\text{a}}=2\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-2\hat{\text{k}}$ and $\vec{\text{c}}=2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}.$
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Answer

projection of $\big(\vec{\text{b}}+\vec{\text{c}}\big)$ on $\vec{\text{a}}$
$=\frac{\big(\vec{\text{b}}+\vec{\text{c}}\big).\vec{\text{a}}}{|\vec{\text{a}}|}$
$=\frac{\vec{\text{b}}.\vec{\text{a}}+\vec{\text{c}}.\vec{\text{a}}}{\sqrt{(2)^2+(-2)^2+(1)^2}}$
$=\frac{\big(\hat{\text{i}}+2\hat{\text{j}}-2\hat{\text{k}}\big)\big(2\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}\big)\big(2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}\big)\big(2\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}\big)}{\sqrt{4+4+1}}$
$=\frac{(1)(2)+(2)(-2)+(-2)(1)+(2)(2)+(-1)(-2)+(4)(1)}{\sqrt{9}}$
$=\frac{2-4-2+4+2+4}{3}$
$=\frac{12-6}{3}=\frac{6}{3}=2$
projection of $\big(\vec{\text{b}}+\vec{\text{c}}\big)=2$

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