Which of the following statements are correct ?
(A) \( \hat{\imath}\times\left(\hat{\jmath}\times\hat{k}\right)=\vec{0} \).
(B) If \( \left|\vec{a}\times\vec{b}\right|=2,\ \vec{a}.\vec{b}=4 \), then \( |\vec{a}|^2\left|\vec{b}\right|^2=6 \).
(C) The number of vectors of unit length perpendicular to \( \vec{a} \) and \( \vec{b} \) are 2.
(D) If \( \vec{a} \) and \( \vec{b} \) represent the diagonals of the rhombus, then \( \vec{a}.\vec{b}=0 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C) and (D) only
For the unit vectors,
\( \hat{\jmath}\times\hat{k}=\hat{\imath} \)
\( \hat{\imath}\times\hat{\imath}=\vec{0} \)
By the identity \( |\vec{a}\times\vec{b}|^2+(\vec{a}\cdot\vec{b})^2=|\vec{a}|^2|\vec{b}|^2 \), with \( |\vec{a}\times\vec{b}|=2 \) and \( \vec{a}\cdot\vec{b}=4 \):
\( |\vec{a}|^2|\vec{b}|^2=4+16 \)
\( =20 \)
The unit vectors perpendicular to both \( \vec{a} \) and \( \vec{b} \) (non-parallel) are
\( \displaystyle \pm\frac{\vec{a}\times\vec{b}}{|\vec{a}\times\vec{b}|} \)
So \( \hat{\imath}\times(\hat{\jmath}\times\hat{k})=\vec{0} \), \( |\vec{a}|^2|\vec{b}|^2=20 \), there are 2 unit vectors perpendicular to both, and the diagonals of a rhombus are perpendicular.
Checking the statements
(A) \( \hat{\imath}\times\left(\hat{\jmath}\times\hat{k}\right)=\vec{0} \). — Correct. \( \hat{\imath}\times\hat{\imath}=\vec{0} \).
(B) If \( \left|\vec{a}\times\vec{b}\right|=2,\ \vec{a}.\vec{b}=4 \), then \( |\vec{a}|^2\left|\vec{b}\right|^2=6 \). — Incorrect. The value is 4 + 16 = 20, not 6.
(C) The number of vectors of unit length perpendicular to \( \vec{a} \) and \( \vec{b} \) are 2. — Correct. They are \( \pm\frac{\vec{a}\times\vec{b}}{|\vec{a}\times\vec{b}|} \).
(D) If \( \vec{a} \) and \( \vec{b} \) represent the diagonals of the rhombus, then \( \vec{a}.\vec{b}=0 \) — Correct. The diagonals of a rhombus are perpendicular.