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: Understanding the Concept:
In parallelogram OBDC the vertex D is opposite O, and C and B are opposite each other. So \(\overline{OD} = \overline{OB} + \overline{OC}\) and \(\overline{CB} = \overline{OB} - \overline{OC}\).

Step 2: Find OB and OC:
\(\overline{OB} = \frac12(\overline{OD} + \overline{CB}) = \frac12(-2\hat i + 2\hat j + 4\hat k) = -\hat i + \hat j + 2\hat k\).
\(\overline{OC} = \frac12(\overline{OD} - \overline{CB}) = \frac12(4\hat i + 2\hat j + 8\hat k) = 2\hat i + \hat j + 4\hat k\).

Step 3: Triple product:
\[ \begin{vmatrix} 1 & 2 & 3 \\ -1 & 1 & 2 \\ 2 & 1 & 4 \end{vmatrix} = 1(4-2) - 2(-4-4) + 3(-1-2) = 2 + 16 - 9 = 9 \]

Step 4: Why the other options are wrong.
Values 3, 6 and 12 arise from sign or arithmetic slips, for example dropping the \(-2(-8)\) term gives \(2 - 9 = -7\).

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