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Question:

Find $|\vec{a} \times \vec{b}|$, if $\vec{a}=\hat{i}-7 \hat{j}+7 \hat{k}$ and $\vec{b}=3 \hat{i}-2 \hat{j}+2 \hat{k}$

Solution:

We have,

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

$\vec{a} \times \vec{b}=\left|\begin{array}{rrc}\hat{i} & \hat{j} & \hat{k} \\ 1 & -7 & 7 \\ 3 & -2 & 2\end{array}\right|$

$=\hat{i}(-14+14)-\hat{j}(2-21)+\hat{k}(-2+21)=19 \hat{j}+19 \hat{k}$

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

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