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Vector or Cross Product — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSVector or Cross Product3 Marks
Question
If $\vec{\text{a}}=3\hat{\text{i}}-\hat{\text{j}}-2\hat{\text{k}}$ and $\vec{\text{b}}=2\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}},$ find $\big(\vec{\text{a}}+2\vec{\text{b}}\big)\times\big(\vec{\text{a}}-2\vec{\text{b}}\big).$
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Answer

Given: $\vec{\text{a}}=3\hat{\text{i}}-\hat{\text{j}}-2\hat{\text{k}}$ $\vec{\text{b}}=2\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}}$$\therefore\vec{\text{a}}+2\vec{\text{b}}=3\hat{\text{i}}-\hat{\text{j}}-2\hat{\text{k}}+2\big(2\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}}\big)$
$=7\hat{\text{i}}+5\hat{\text{j}}+0\hat{\text{k}}$
$\therefore2\vec{\text{a}}-\vec{\text{b}}\big(3\hat{\text{i}}-\hat{\text{j}}-2\hat{\text{k}}\big)-\big(2\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}}\big)$
$=4\hat{\text{i}}-5\hat{\text{j}}-5\hat{\text{k}}$
$\big(\vec{\text{a}}+2\vec{\text{b}}\big)\times\big(2\vec{\text{a}}-\vec{\text{b}}\big)=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\7&5&0\\4&-5&-5 \end{vmatrix}$
$=\hat{\text{i}}(-25+0)-\hat{\text{j}}(-35+0)+\hat{\text{k}}(-35-20)$
$=-25\hat{\text{i}}+35\hat{\text{j}}-55\hat{\text{k}}$

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