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Mathematics Question on Vector Algebra

Find ∣a⃗×b⃗∣|\vec{a}\times\vec{b}| if a⃗=i^−7j^+7k^  and  b⃗=3i^−2j^+2k^.\vec{a}=\hat{i}-7\hat{j}+7\hat{k}\space and\space \vec{b}=3\hat{i}-2\hat{j}+2\hat{k}.

Answer

We have,
a⃗=i^−7j^+7k^\vec{a}=\hat{i}-7\hat{j}+7\hat{k}   and  b⃗=3i^−2j^+2k^.\space and\space \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}
=i^(−14+14)−j^(2−21)+k^(−2+21)=19j^+19k^=\hat{i}(-14+14)-\hat{j}(2-21)+\hat{k}(-2+21)=19\hat{j}+19\hat{k}
∴∣a⃗×b⃗∣=(19)2+(19)2=2×(19)2∴|\vec{a}\times\vec{b}|=\sqrt{(19)^{2}+(19)^{2}}=\sqrt{2×(19)^{2}}
=192=19\sqrt{2}