Which of the following statements are correct?
(A) \(\vec{a}\times\vec{b} = -(\vec{b}\times\vec{a})\) for any two vectors \(\vec{a}\) and \(\vec{b}\)
(B) \(\hat{\imath}\times\hat{k} = \hat{\jmath}\)
(C) \(|\vec{a}\times\vec{b}| = |\vec{a}|\,|\vec{b}|\sin\theta\), where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\)
(D) If \(\vec{a}\) and \(\vec{b}\) are non-zero parallel vectors, then \(\vec{a}\times\vec{b} = \vec{0}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (C) and (D) only
The vector product is \(\vec{a}\times\vec{b} = |\vec{a}|\,|\vec{b}|\sin\theta\,\hat{n}\), where \(\hat{n}\) is the unit vector perpendicular to both, given by the right-hand rule. Reversing the order reverses \(\hat{n}\).
So \(\vec{a}\times\vec{b} = -(\vec{b}\times\vec{a})\), \(|\vec{a}\times\vec{b}| = |\vec{a}|\,|\vec{b}|\sin\theta\), parallel vectors give \(\vec{0}\), and \(\hat{\imath}\times\hat{k} = -\hat{\jmath}\).
Checking the statements
(A) \(\vec{a}\times\vec{b} = -(\vec{b}\times\vec{a})\) — Correct. Swapping the order reverses the direction \(\hat{n}\).
(B) \(\hat{\imath}\times\hat{k} = \hat{\jmath}\) — Incorrect. \(\hat{k}\times\hat{\imath} = \hat{\jmath}\), so \(\hat{\imath}\times\hat{k} = -\hat{\jmath}\).
(C) \(|\vec{a}\times\vec{b}| = |\vec{a}|\,|\vec{b}|\sin\theta\) — Correct. \(\hat{n}\) is a unit vector and \(\sin\theta \ge 0\) for \(0 \le \theta \le \pi\).
(D) Non-zero parallel vectors give \(\vec{a}\times\vec{b} = \vec{0}\) — Correct. \(\theta = 0\) or \(\pi\), so \(\sin\theta = 0\).