Match List-I with List-II
| List-I | List-II | ||
| (A) | Area of the parallelogram with adjacent sides \(\hat{\imath}+\hat{\jmath}\) and \(\hat{\imath}-\hat{\jmath}\) | (I) | \(\displaystyle\frac{\sqrt{3}}{2}\) |
| (B) | Area of the triangle with two sides given by the vectors \(2\hat{\imath}\) and \(3\hat{k}\) | (II) | 4 |
| (C) | Area of the triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) | (III) | 2 |
| (D) | \(|\vec{a}\times\vec{b}|\), where \(|\vec{a}| = 2\), \(|\vec{b}| = 4\) and the angle between \(\vec{a}\) and \(\vec{b}\) is \(\displaystyle\frac{\pi}{6}\) | (IV) | 3 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
(A) Parallelogram with adjacent sides \(\hat{\imath}+\hat{\jmath}\) and \(\hat{\imath}-\hat{\jmath}\)
This matches (III) 2.
\((\hat{\imath}+\hat{\jmath})\times(\hat{\imath}-\hat{\jmath}) = -\hat{\imath}\times\hat{\jmath}+\hat{\jmath}\times\hat{\imath} = -2\hat{k}\)
The area of the parallelogram is \(|-2\hat{k}| = 2\).
(B) Triangle with two sides \(2\hat{\imath}\) and \(3\hat{k}\)
This matches (IV) 3.
\(2\hat{\imath}\times 3\hat{k} = 6(\hat{\imath}\times\hat{k}) = -6\hat{\jmath}\)
The area of the triangle is half the magnitude: \(\frac{1}{2}\times 6 = 3\).
(C) Triangle with vertices A(1, 0, 0), B(0, 1, 0), C(0, 0, 1)
This matches (I) \(\frac{\sqrt{3}}{2}\).
\(\overrightarrow{AB} = -\hat{\imath}+\hat{\jmath}, \quad \overrightarrow{AC} = -\hat{\imath}+\hat{k}\)
\( \displaystyle \overrightarrow{AB}\times\overrightarrow{AC} = \begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{vmatrix} = \hat{\imath}+\hat{\jmath}+\hat{k} \)
\( \displaystyle \text{Area} = \frac{1}{2}\sqrt{1+1+1} = \frac{\sqrt{3}}{2} \)
(D) \(|\vec{a}\times\vec{b}|\) with \(|\vec{a}| = 2\), \(|\vec{b}| = 4\), angle \(\frac{\pi}{6}\)
This matches (II) 4.
\( \displaystyle |\vec{a}\times\vec{b}| = 2\times 4\times\sin\frac{\pi}{6} = 8\times\frac{1}{2} = 4 \)