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If $\mathbf{a}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}, \quad \mathbf{b}=3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ and $\mathbf{c}=5 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$, then the volume of the parallelopiped with coterminous edges $\mathbf{a}+\mathbf{b}, \mathbf{b}+\mathbf{c}, \mathbf{c}+\mathbf{a}$, is
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1243 Upvotes
Verified Answer
The correct answer is:
16
We have,
$\begin{aligned}
& \mathbf{a}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}} \\
& \mathbf{b}=3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}} \text { and } \mathbf{c}=5 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}
\end{aligned}$
Now, $\mathbf{a}+\mathbf{b}=5 \hat{\mathbf{i}}-7 \hat{\mathbf{j}}+10 \hat{\mathbf{k}}$ and $\mathbf{b}+\mathbf{c}=8 \hat{\mathbf{i}}-7 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $\mathbf{c}+\mathbf{a}=7 \hat{\mathbf{i}}-6 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$
$\therefore$ Volume of the parallelopiped with coterminous edges $\mathbf{a}+\mathbf{b}, \mathbf{b}+\mathbf{c}, \mathbf{c}+\mathbf{a}=[\mathbf{a}+\mathbf{b} \mathbf{b}+\mathbf{c} \mathbf{c}+\mathbf{a}]$
$\begin{aligned}
& =\left|\begin{array}{ccc}
5 & -7 & 10 \\
8 & -7 & 3 \\
7 & -6 & 3
\end{array}\right| \\
& =5(-21+18)+7(24-21)+10(-48+49) \\
& =-15+21+10=16
\end{aligned}$
$\begin{aligned}
& \mathbf{a}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}} \\
& \mathbf{b}=3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}} \text { and } \mathbf{c}=5 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}
\end{aligned}$
Now, $\mathbf{a}+\mathbf{b}=5 \hat{\mathbf{i}}-7 \hat{\mathbf{j}}+10 \hat{\mathbf{k}}$ and $\mathbf{b}+\mathbf{c}=8 \hat{\mathbf{i}}-7 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $\mathbf{c}+\mathbf{a}=7 \hat{\mathbf{i}}-6 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$
$\therefore$ Volume of the parallelopiped with coterminous edges $\mathbf{a}+\mathbf{b}, \mathbf{b}+\mathbf{c}, \mathbf{c}+\mathbf{a}=[\mathbf{a}+\mathbf{b} \mathbf{b}+\mathbf{c} \mathbf{c}+\mathbf{a}]$
$\begin{aligned}
& =\left|\begin{array}{ccc}
5 & -7 & 10 \\
8 & -7 & 3 \\
7 & -6 & 3
\end{array}\right| \\
& =5(-21+18)+7(24-21)+10(-48+49) \\
& =-15+21+10=16
\end{aligned}$
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