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Question
If \[\vec{a} = 2\hat{ i} - 3 \hat { j} + 5 \hat { k} , \vec{b} = 3 \hat {i} - 4 \hat {j} + 5 \hat {k} \text { and } \vec{c} = 5\hat { i } - 3 \hat {j}- 2 \hat{k},\] then the volume of the parallelopiped with conterminous edges \[\vec{a} + \vec{b,} \vec{b} + \vec{c,} \vec{c} + \vec{a}\] is
Options
2
1
-1
0
None of these
Solution
We have
\[ \vec{a} + \vec{b} = \left( 2 \hat {i} - 3 \hat {j} + 5 \hat { k} \right) + \left( 3 \hat {i} - 4 \hat {j} + 5 \hat {k} \right) = 5 \hat {i} - 7 \hat {j} + 10 \hat {k} \]
\[ \vec{b} + \vec{c} = \left( 3 \hat {i} - 4 \hat {j} + 5 \hat {k} \right) + \left( 5 \hat {i} - 3 \hat {j} - 2 \hat {k} \right) = 8 \hat {i} - 7 \hat {j} + 3 \hat {k } \]
\[ \vec{c} + \vec{a} = \left( 5\hat { i } - 3 \hat {j} - 2 \hat {k} \right) + \left( 2 \hat {i} - 3 \hat {j} + 5 \hat {k} \right) = 7 \hat {i} - 6 \hat {j} + 3 \hat {k} \]
\[\text { We know that the volume of a parallelopiped whose three adjacent adges are } \vec{a} + \vec{b} , \vec{b} + \vec{c} \text { and } \vec{c} + \vec{a} \text { is equal to } \left| \left[ \vec{a} + \vec{b} \vec{b} + \vec{c} \vec{c} + \vec{a} \right] \right| . \]
We have
\[\left[ \vec{a} + \vec{b} \vec{b} + \vec{c} \vec{c} + \vec{a} \right] = \begin{vmatrix}5 & - 7 & 10 \\ 8 & - 7 & 3 \\ 7 & - 6 & 3\end{vmatrix}\]
\[ \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} = 5\left( - 21 + 18 \right) + 7\left( 24 - 21 \right) + 10\left( - 48 + 49 \right)\]
\[ = \left( 5 \times - 3 \right) + \left( 7 \times 3 \right) + \left( 10 \times 1 \right)\]
\[ = 16\]
\[ \therefore \text { Volume of parallelopiped } = \left| 16 \right| = 16\]
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