Question

Let \(\overline{OD} = \hat{i}+2\hat{j}+6\hat{k}\), \(\overline{CB} = -3\hat{i}-2\hat{k}\) be the diagonals of the parallelogram OBDC and \(\overline{OA} = \hat{i}+2\hat{j}+3\hat{k}\) be another vector. Then the volume of a parallelopiped determined by vectors \(\overline{OA}\), \(\overline{OB}\), and \(\overline{OC}\) (in cubic units), is

• A. \(3\)
• B. \(6\)
• C. \(9\)
• D. \(12\)

Hint:

Diagonals of parallelogram OBDC give OB and OC by sum and difference.

Answer 1

The Correct Option is C

Step 1: Midpoint idea:
The diagonals of a parallelogram bisect each other. So $\overline{OB}$ and $\overline{OC}$ are the half-sum and half-difference of the diagonal vectors, in the right order.

Step 2: Compute:
$\overline{OB} = (-1, 1, 2)$ and $\overline{OC} = (2, 1, 4)$.

Step 3: Expand the determinant along the first row:
$1\cdot(1\cdot4 - 2\cdot1) - 2\cdot((-1)\cdot4 - 2\cdot2) + 3\cdot((-1)\cdot1 - 1\cdot2) = 2 + 16 - 9 = 9$. The volume is the absolute value, 9.

Final Answer:
The volume is 9, option (C). \[ \boxed{9\text{ cubic units}} \]