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Scalar Triple Product — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsScalar Triple Product3 Marks
Question
Find $\Big[\vec{\text{a}}\ \vec{\text{b}}\ \vec{\text{c}}\Big]$, when
$\vec{\text{a}}=\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}},\vec{\text{b}}=2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}$ and $\vec{\text{c}}=\hat{\text{j}}+\hat{\text{k}}$
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Answer

Given:
$\vec{\text{a}}=\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}}$
$\vec{\text{b}}=2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}$
$\vec{\text{c}}=\hat{\text{j}}+\hat{\text{k}}$
$\therefore\vec{\text{a}}\times\vec{\text{b}}=\big(\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}}\big)\times\big(2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}\big)$
$=\hat{\text{k}}+\hat{\text{j}}+4\hat{\text{k}}+2\hat{\text{i}}+6\hat{\text{j}}-3\hat{\text{i}}$
$=-\hat{\text{i}}+7\hat{\text{j}}+5\hat{\text{k}}$
$\big(\vec{\text{a}}\times\vec{\text{b}}\big).\vec{\text{c}}=\big(-\hat{\text{i}}+7\hat{\text{j}}+5\hat{\text{k}}\big).\big(\hat{\text{j}}+\hat{\text{k}}\big)$
$=7+5=12\ ...(1)$
Now,
$\big[\vec{\text{a}}\ \vec{\text{b}}\ \vec{\text{c}}\big]=\big(\vec{\text{a}}\times\vec{\text{b}}\big).\vec{\text{c}}$
$=12$ [Using (1)]

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