Math, asked by talpadadilip417, 28 days ago


 \boxed{\boxed { \red{ \pmb{ \text{if \:  \:}  \tt\vec{a}=\hat{i}-7 \hat{j}+7 \hat{k}  \text{\:  \: and \:  \: } \tt\vec{b}=3\hat{i}-2\hat{j}+2\hat{k} \: \:  find \:  \:  |\vec{a} \times  \vec{b}|. }}}}
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Answered by StarFighter
11

Answer:

Given :-

\bigstar \: \: \sf \vec{a} =\: \hat{i} - 7 \hat{j} + 7 \hat{k}

\bigstar \: \: \sf \vec{b} =\: 3\hat{i} - 2\hat{j} + 2\hat{k}

To Find :-

\bigstar \: \: \sf |\vec{a} \times \vec{b}|

Solution :-

\mapsto \sf \vec{a} =\: \hat{i} - 7\hat{j} + 7\hat{k}

\longrightarrow \bf \vec{a} =\: 1\hat{i} - 7\hat{j} + 7\hat{k}

And,

\longrightarrow \bf \vec{b} =\: 3\hat{i} - 2\hat{j} + 2\hat{k}

Now,

\implies \sf \vec{a} \times \vec{b} =\: \begin{array}{|ccc|} \hat{i} & \hat{j} & \hat{k} \\ 1 &- 7 &7 \\ 3 &- 2&2\end{array}

\implies \sf \vec{a} \times \vec{b} =\: \hat{i}\{(- 7 \times 2) - (- 2 \times 7)\} - \hat{j}\{(1 \times 2) - (3 \times 7)\} + \hat{k}\{(1 \times - 2) - (3 \times - 7)\}\\

\implies \sf \vec{a} \times \vec{b} =\: \hat{i}\{- 14 - (- 14)\} - \hat{j}\{2 - 21\} + \hat{k}\{- 2 - (- 21)\}\\

\implies \sf \vec{a} \times \vec{b} =\: \hat{i}(- 14 + 14) - \hat{j}(- 19) + \hat{k}(- 2 + 21)\\

\implies \sf \vec{a} \times \vec{b} =\: \hat{i}(0) - \hat{j}(- 19) + \hat{k}(19)\\

\implies \sf\bold{\purple{\vec{a} \times \vec{b} =\: 0\hat{i} + 19\hat{j} + 19\hat{k}}}

Now,

\dashrightarrow \bf |\vec{a} \times \vec{b}| =\: \sqrt{\bigg(0\bigg)^2 + \bigg(19\bigg)^2 + \bigg(19\bigg)^2}\\

\dashrightarrow \sf |\vec{a} \times \vec{b}| =\: \sqrt{\bigg(0 \times 0\bigg) + \bigg(19 \times 19\bigg) + \bigg(19 \times 19\bigg)}\\

\dashrightarrow \sf |\vec{a} \times \vec{b}| =\: \sqrt{0 + 361 + 361}\\

\dashrightarrow \sf |\vec{a} \times \vec{b}| =\: \sqrt{722}\\

\dashrightarrow \sf\bold{\red{|\vec{a} \times \vec{b}| =\: 19\sqrt{2}}}\\

\sf\boxed{\bold{\pink{\therefore\: The\: value\: of\: |\vec{a} \times \vec{b}|\: is\: 19\sqrt{2}\: .}}}\\

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