A cube of side 2 has one vertex at the origin O and three edges along the positive coordinate axes. P(2, 2, 2), A(2, 2, 0) and B(0, 2, 2) are vertices of the cube.
Match List-I with List-II
| List-I | List-II | ||
| (A) | \(|\overrightarrow{OP}|\) | (I) | 4 |
| (B) | \(|\overrightarrow{OA}|\) | (II) | \(2\sqrt{2}\) |
| (C) | \(\overrightarrow{OA}\cdot\overrightarrow{OB}\) | (III) | \(4\sqrt{3}\) |
| (D) | \(|\overrightarrow{OA}\times\overrightarrow{OB}|\) | (IV) | \(2\sqrt{3}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(IV), (B)-(II), (C)-(I), (D)-(III)
With O at the origin, the position vectors are
\(\overrightarrow{OP} = 2\hat{\imath}+2\hat{\jmath}+2\hat{k}, \quad \overrightarrow{OA} = 2\hat{\imath}+2\hat{\jmath}, \quad \overrightarrow{OB} = 2\hat{\jmath}+2\hat{k}\)
(A) \(|\overrightarrow{OP}|\), the body diagonal
This matches (IV) \(2\sqrt{3}\).
\(|\overrightarrow{OP}| = \sqrt{4+4+4} = 2\sqrt{3}\)
(B) \(|\overrightarrow{OA}|\), a face diagonal
This matches (II) \(2\sqrt{2}\).
\(|\overrightarrow{OA}| = \sqrt{4+4} = 2\sqrt{2}\)
(C) \(\overrightarrow{OA}\cdot\overrightarrow{OB}\)
This matches (I) 4.
\(\overrightarrow{OA}\cdot\overrightarrow{OB} = (2)(0)+(2)(2)+(0)(2) = 4\)
(D) \(|\overrightarrow{OA}\times\overrightarrow{OB}|\)
This matches (III) \(4\sqrt{3}\).
\(\overrightarrow{OA}\times\overrightarrow{OB} = (2\cdot 2-0\cdot 2)\hat{\imath}-(2\cdot 2-0\cdot 0)\hat{\jmath}+(2\cdot 2-2\cdot 0)\hat{k} = 4\hat{\imath}-4\hat{\jmath}+4\hat{k}\)
\(|\overrightarrow{OA}\times\overrightarrow{OB}| = \sqrt{16+16+16} = 4\sqrt{3}\)