If \(\vec{a}\) and \(\vec{b}\) are two non zero vectors, then which of the following is correct:
A. \(\vec{a}\times\vec{b}=\vec{0}\) if and only if \(\vec{a}\) is perpendicular to \(\vec{b}\).
B. \(\vec{a}\times\vec{b}\) is vector.
C. \(\vec{a}\times\vec{b}=|\vec{a}||\vec{b}|\cos\theta\,\hat{n}\)
D. If \(\theta=\displaystyle\frac{\pi}{2}\), \(|\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → B and D only
The vector (cross) product of \(\vec{a}\) and \(\vec{b}\), with angle \(\theta\) between them, is defined as
\(\vec{a}\times\vec{b} = |\vec{a}||\vec{b}|\sin\theta\,\hat{n}\)
where \(\hat{n}\) is the unit vector perpendicular to both \(\vec{a}\) and \(\vec{b}\). Its magnitude is
\(|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta\)
For non-zero vectors this is 0 only when \(\sin\theta = 0\), that is, \(\theta = 0\) or \(\pi\).
So the cross product is a vector with sine (not cosine) in it, and it is zero exactly when the vectors are parallel.
Checking the statements
(A) \(\vec{a}\times\vec{b}=\vec{0}\) if and only if \(\vec{a}\) is perpendicular to \(\vec{b}\). — Incorrect. It is zero when the vectors are parallel; for perpendicular vectors \(\sin\theta = 1\).
(B) \(\vec{a}\times\vec{b}\) is vector. — Correct. It has magnitude \(|\vec{a}||\vec{b}|\sin\theta\) and direction \(\hat{n}\).
(C) \(\vec{a}\times\vec{b}=|\vec{a}||\vec{b}|\cos\theta\,\hat{n}\) — Incorrect. The definition has \(\sin\theta\), not \(\cos\theta\).
(D) If \(\theta=\frac{\pi}{2}\), \(|\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|\) — Correct. \(\sin\frac{\pi}{2} = 1\).