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Scalar Triple Product — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsScalar Triple Product2 Marks
Question
Find $\vec{\text{a}}.\big(\vec{\text{b}}\times\vec{\text{c}}\big), $ if $\vec{\text{a}}=2\hat{\text{i}}+\hat{\text{j}}+3\hat{\text{k}},\vec{\text{b}}=-\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{c}}=3\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}$.
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Answer

The given vectors are $\vec{\text{a}}=2\hat{\text{i}}+\hat{\text{j}}+3\vec{\text{k}},\vec{\text{b}}=-\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{c}}=3\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}$
Now,
$\vec{\text{b}}\times\vec{\text{c}}=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\-1&2&1\\3&1&2 \end{vmatrix}=3\hat{\text{i}}+5\hat{\text{j}}-7\hat{\text{k}}$
$\therefore\vec{\text{a}}\big(\vec{\text{b}}\times\vec{\text{c}}\big)=\big(2\hat{\text{i}}+\hat{\text{j}}+3\hat{\text{k}}\big).\big(3\hat{\text{i}}+5\hat{\text{j}}-7\hat{\text{k}}\big)$
$=2\times3+1\times5=3\times(-7)$
$=6+5-21=-10$

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