Let \( \vec{a}=\hat{\imath}+\hat{\jmath} \) and \( \vec{b}=\hat{\imath}-\hat{\jmath} \), (where \( \mu,\lambda \) are scalars) then
Match List-I with List-II
| List-I | List-II | ||
| (A) | vector \( 2\vec{a}+2\vec{b} \) is | (I) | \( \mu\hat{\jmath} \) |
| (B) | unit vector perpendicular to both \( \vec{a}+\vec{b} \) and \( \vec{a}-\vec{b} \) | (II) | \( \lambda\hat{\imath} \) |
| (C) | Any vector parallel to \( \vec{a}+\vec{b} \) | (III) | \( 4\hat{\imath} \) |
| (D) | Any vector parallel to \( \vec{a}-\vec{b} \) | (IV) | \( \hat{k} \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) - (III), (B) - (IV), (C) - (II), (D) - (I)
Here \( \vec{a}+\vec{b}=2\hat{\imath} \) and \( \vec{a}-\vec{b}=2\hat{\jmath} \).
(A) vector \( 2\vec{a}+2\vec{b} \)
This matches (III) \( 4\hat{\imath} \).
\( 2\vec{a}+2\vec{b}=2(2\hat{\imath})=4\hat{\imath} \).
(B) unit vector perpendicular to both \( \vec{a}+\vec{b} \) and \( \vec{a}-\vec{b} \)
This matches (IV) \( \hat{k} \).
A vector perpendicular to both \( \hat{\imath} \) and \( \hat{\jmath} \) is along \( \hat{\imath}\times\hat{\jmath}=\hat{k} \), which is already a unit vector.
(C) Any vector parallel to \( \vec{a}+\vec{b} \)
This matches (II) \( \lambda\hat{\imath} \).
Any vector parallel to \( 2\hat{\imath} \) is a scalar multiple \( \lambda\hat{\imath} \).
(D) Any vector parallel to \( \vec{a}-\vec{b} \)
This matches (I) \( \mu\hat{\jmath} \).
Any vector parallel to \( 2\hat{\jmath} \) is a scalar multiple \( \mu\hat{\jmath} \).
So the matching is (A)-(III), (B)-(IV), (C)-(II), (D)-(I).