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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA2 Marks
Question
$\text{Find}\ |\vec{a}\ \times\vec{b}|,\ \text{if}\ \vec{a}=\hat{i}-7\hat{j}+7\hat{k}\ \text{and}\ \vec{b}=3\hat{i}-2\hat{j}+2\hat{k}.$
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Answer

$\text{Give:}\ \ \vec{a}=\hat{i}-7\hat{j}+7\hat{k}\ \text{and}\ \vec{b}=3\hat{i}-2\hat{j}+2\hat{k}$

$\vec{a}\times\vec{b}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\1&-7&7\\3&-2&2\end{vmatrix}$

Expanding along first row,

$\vec{a}\times\vec{b}=\hat{i}\begin{vmatrix}-7&7\\-2&2\end{vmatrix}-\hat{j}\begin{vmatrix}1&7\\3&2\end{vmatrix}+\hat{k}\begin{vmatrix}1&-7\\3&-2\end{vmatrix}$ $=\hat{i}(-14+14)-\hat{j}(2-21)+\hat{k}(-2+21)$

$=0\hat{i}+19\hat{j}+19\hat{k}$

$\therefore\ \ \Big|\vec{a}\times\vec{b}\Big|=\sqrt{(0)^2+(19)^2+(19)^2}=\sqrt{2(19)^2}=19\sqrt{2}$

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