Match List-I with List-II.
| List-I | List-II | ||
| (A) | Area of triangle Δ with adjacent sides \(\vec{a}\) and \(\vec{b}\) | (I) | \(\vec{a}\times\vec{b}\) |
| (B) | Area of parallelogram with adjacent side \(\vec{a}\times\vec{b}\) | (II) | \(\displaystyle\frac{1}{2}|\vec{a}\times\vec{b}|\) |
| (C) | \((\vec{a}-\vec{b})\times(\vec{a}+\vec{b})\) | (III) | \(|\vec{a}\times\vec{b}|\) |
| (D) | \(|\vec{a}||\vec{b}|\sin\theta\,\hat{n}\), where symbols have their usual meaning | (IV) | \(2(\vec{a}\times\vec{b})\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(II), (B)-(III), (C)-(IV), (D)-(I)
For vectors \(\vec{a}\) and \(\vec{b}\) with angle \(\theta\) between them, the vector product is
\( \vec{a}\times\vec{b}=|\vec{a}||\vec{b}|\sin\theta\,\hat{n} \)
and its magnitude \(|\vec{a}||\vec{b}|\sin\theta\) is the area of the parallelogram with adjacent sides \(\vec{a}\) and \(\vec{b}\).
(A) Area of triangle Δ with adjacent sides \(\vec{a}\) and \(\vec{b}\)
This matches (II) \(\frac{1}{2}|\vec{a}\times\vec{b}|\). The triangle is half of the parallelogram on the same two sides.
(B) Area of parallelogram with adjacent side \(\vec{a}\times\vec{b}\)
This matches (III) \(|\vec{a}\times\vec{b}|\), by elimination. As printed, \(\vec{a}\times\vec{b}\) is a single vector, so it cannot give the two adjacent sides of a parallelogram. With adjacent sides \(\vec{a}\) and \(\vec{b}\), the area is \(|\vec{a}\times\vec{b}|\).
(C) \((\vec{a}-\vec{b})\times(\vec{a}+\vec{b})\)
This matches (IV) \(2(\vec{a}\times\vec{b})\). Expanding, with \(\vec{a}\times\vec{a}=\vec{b}\times\vec{b}=\vec{0}\) and \(\vec{b}\times\vec{a}=-\vec{a}\times\vec{b}\):
\( (\vec{a}-\vec{b})\times(\vec{a}+\vec{b})=\vec{a}\times\vec{a}+\vec{a}\times\vec{b}-\vec{b}\times\vec{a}-\vec{b}\times\vec{b} \)
\( =\vec{0}+\vec{a}\times\vec{b}+\vec{a}\times\vec{b}-\vec{0} \)
\( =2(\vec{a}\times\vec{b}) \)
(D) \(|\vec{a}||\vec{b}|\sin\theta\,\hat{n}\)
This matches (I) \(\vec{a}\times\vec{b}\). It is the definition of the vector product.
Note: It seems that there is a mistake in the original NTA paper, which prints List-I item (B) as “Area of parallelogram with adjacent side \(\vec{a}\times\vec{b}\)” when it should read “Area of parallelogram with adjacent sides \(\vec{a}\) and \(\vec{b}\)”. The question can still be solved because (A) and (C) match only (II) and (IV), and Option (3) is the only option with (A)-(II) and (C)-(IV).