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Vector or Cross Product — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector or Cross Product2 Marks
Question
If $\vec{\text{a}}=\hat{\text{i}}+3\hat{\text{j}}-2\hat{\text{k}}$ and $\vec{\text{b}}=-\hat{\text{i}}+3\hat{\text{k},}$ find $\big|\vec{\text{a}}\times\vec{\text{b}}\big|.$
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Answer

If $\vec{\text{a}}=\text{a}_1\hat{\text{i}}+\text{b}_1\hat{\text{j}}+\text{c}_1\hat{\text{k}}$ and
$\vec{\text{b}}=\text{a}_2\hat{\text{i}}+\text{b}_2\hat{\text{j}}+\text{c}_2\hat{\text{k}},$ then
$\vec{\text{a}}\times\vec{\text{b}}=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\\text{a}_1&\text{b}_1&\text{c}_2\\\text{a}_2&\text{b}_2&\text{c}_2 \end{vmatrix}$
$=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\1&3&-2\\-1&0&3 \end{vmatrix}$
$=\hat{\text{i}}(9-0)-\hat{\text{j}}(3-2)+\hat{\text{k}}(0+3)$
$\vec{\text{a}}\times\vec{\text{b}}=9\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$
$\big|\vec{\text{a}}\times\vec{\text{b}}\big|=\sqrt{(9)^2+(-1)^2+(3)^2}$
$=\sqrt{81+1+9}$
$\big|\vec{\text{a}}\times\vec{\text{b}}\big|=\sqrt{91}$

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