Question:
The volume of the parallelopiped whose coterminous edges are \(2\hat{i}+\hat{j}-\hat{k},3\hat{i}-\hat{j}-\hat{k},\hat{j}+3\hat{k}\) is
The volume of the parallelopiped whose coterminous edges are \(2\hat{i}+\hat{j}-\hat{k},3\hat{i}-\hat{j}-\hat{k},\hat{j}+3\hat{k}\) is
Show Hint
The volume of a parallelepiped equals the absolute value of the determinant of its edge vectors.
Updated On: Oct 1, 2026
- \(16\) cu. units
- \(6\) cu. units
- \(2\) cu. units
- \(12\) cu. units
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept
The volume of a parallelepiped with coterminous edges \(\vec a,\vec b,\vec c\) is \(|[\vec a\ \vec b\ \vec c]|\), the absolute value of the determinant of their components.
Step 2: Determinant
\[ \begin{vmatrix}2&1&-1\\3&-1&-1\\0&1&3\end{vmatrix} \]
Expand along the first row:
\[ 2\big((-1)(3)-(-1)(1)\big)-1\big(3\cdot3-(-1)\cdot0\big)+(-1)\big(3\cdot1-(-1)\cdot0\big) \]
\[ =2(-3+1)-9-3=-4-9-3=-16 \]
Step 3: Volume
\[ V=|-16|=16\ \text{cubic units} \]
This is option (A).
Final Answer:
The determinant of the three edges is -16, so the volume is 16 cubic units, option (A). \[ \boxed{16\ \text{cu. units}} \]
The volume of a parallelepiped with coterminous edges \(\vec a,\vec b,\vec c\) is \(|[\vec a\ \vec b\ \vec c]|\), the absolute value of the determinant of their components.
Step 2: Determinant
\[ \begin{vmatrix}2&1&-1\\3&-1&-1\\0&1&3\end{vmatrix} \]
Expand along the first row:
\[ 2\big((-1)(3)-(-1)(1)\big)-1\big(3\cdot3-(-1)\cdot0\big)+(-1)\big(3\cdot1-(-1)\cdot0\big) \]
\[ =2(-3+1)-9-3=-4-9-3=-16 \]
Step 3: Volume
\[ V=|-16|=16\ \text{cubic units} \]
This is option (A).
Final Answer:
The determinant of the three edges is -16, so the volume is 16 cubic units, option (A). \[ \boxed{16\ \text{cu. units}} \]
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