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

• A. \(16\) cu. units
• B. \(6\) cu. units
• C. \(2\) cu. units
• D. \(12\) cu. units

Hint:

The volume of a parallelepiped equals the absolute value of the determinant of its edge vectors.

Answer 1

The Correct Option is A

Step 1: Approach
Compute the cross product of two edges and dot it with the third.

Step 2: Cross product
$\vec b\times\vec c=(3,-1,-1)\times(0,1,3)=((-1)(3)-(-1)(1),\ (-1)(0)-3\cdot3,\ 3\cdot1-0)=(-2,-9,3)$.

Step 3: Dot with a
$(2,1,-1)\cdot(-2,-9,3)=-4-9-3=-16$.

Step 4: Volume
The magnitude is 16 cubic units, option (A).

Final Answer:
The determinant of the three edges is -16, so the volume is 16 cubic units, option (A). \[ \boxed{16\ \text{cu. units}} \]