Let \(\overrightarrow{OA} = 2\hat{\imath} - \hat{\jmath} + \hat{k}\) and \(\overrightarrow{OB} = \hat{\imath} + \hat{\jmath} - \hat{k}\). Then
A. The magnitude of vector \(\overrightarrow{OA}\) is 6
B. The magnitude of vector \(\overrightarrow{OB}\) is \(\sqrt{3}\)
C. The vector \(\overrightarrow{AB}\) is \(\left(-\hat{\imath} + 2\hat{\jmath} - 2\hat{k}\right)\)
D. \(\overrightarrow{OA}\cdot\overrightarrow{OB} = 0\)
E. \(\overrightarrow{OA} \parallel \overrightarrow{OB}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → B, C, D Only
With \(\overrightarrow{OA} = (2, -1, 1)\) and \(\overrightarrow{OB} = (1, 1, -1)\):
\(|\overrightarrow{OA}| = \sqrt{4 + 1 + 1} = \sqrt{6}\)
\(|\overrightarrow{OB}| = \sqrt{1 + 1 + 1} = \sqrt{3}\)
\(\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = (-1, 2, -2)\)
\(\overrightarrow{OA}\cdot\overrightarrow{OB} = 2 - 1 - 1 = 0\)
So \(|\overrightarrow{OA}| = \sqrt{6}\), \(|\overrightarrow{OB}| = \sqrt{3}\), \(\overrightarrow{AB} = -\hat{\imath} + 2\hat{\jmath} - 2\hat{k}\) and the dot product is 0, which makes the vectors perpendicular.
Checking the statements
(A) The magnitude of vector \(\overrightarrow{OA}\) is 6 — Incorrect. The magnitude is \(\sqrt{6}\), not 6.
(B) The magnitude of vector \(\overrightarrow{OB}\) is \(\sqrt{3}\) — Correct. \(\sqrt{1 + 1 + 1} = \sqrt{3}\).
(C) The vector \(\overrightarrow{AB}\) is \(\left(-\hat{\imath} + 2\hat{\jmath} - 2\hat{k}\right)\) — Correct. \(\overrightarrow{OB} - \overrightarrow{OA} = (1 - 2,\ 1 + 1,\ -1 - 1)\).
(D) \(\overrightarrow{OA}\cdot\overrightarrow{OB} = 0\) — Correct. 2 − 1 − 1 = 0.
(E) \(\overrightarrow{OA} \parallel \overrightarrow{OB}\) — Incorrect. The components are not proportional (2 : 1 but −1 : 1); the vectors are in fact perpendicular.