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Scalar Or Dot Product — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsScalar Or Dot Product2 Marks
Question
If $\vec{\text{a}}=\hat{\text{i}}-\hat{\text{j}}$ and $\vec{\text{b}}=-\hat{\text{j}}+2\hat{\text{k}},$ find $\big(\vec{\text{a}}-2\vec{\text{b}}\big).\big(\vec{\text{a}}+\vec{\text{b}}\big).$
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Answer

We have
$\vec{\text{a}}=\hat{\text{i}}-\hat{\text{j}}$ and $\vec{\text{b}}=-\hat{\text{j}}+2\hat{\text{k}}$
$\vec{\text{a}}-2\vec{\text{b}}=\big(\hat{\text{i}}-\hat{\text{j}}\big)-2\big(-\hat{\text{j}}+2\hat{\text{k}}\big)\\=\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{j}}-4\hat{\text{k}}=\hat{\text{i}}+\hat{\text{j}}-4\hat{\text{k}}$
$\vec{\text{a}}+\vec{\text{b}}=\hat{\text{i}}-\hat{\text{j}}-\hat{\text{j}}+2\hat{\text{k}}=\hat{\text{i}}-2\hat{\text{j}}+2\hat{\text{k}}$
$\big(\vec{\text{a}}-2\vec{\text{b}}\big).\big(\vec{\text{a}}+\vec{\text{b}}\big)$
$=\big(\hat{\text{i}}+\hat{\text{j}}-4\hat{\text{k}}\big).\big(\hat{\text{i}}-2\hat{\text{j}}+2\hat{\text{k}}\big)$
$=1-2-8$
$=-9$

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