Let \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) be mutually perpendicular vectors with \(|\vec{a}| = 2\), \(|\vec{b}| = 1\) and \(|\vec{c}| = 2\).
Match List-I with List-II
| List-I | List-II | ||
| (A) | \(|\vec{a}+\vec{b}|\) | (I) | 3 |
| (B) | \(|\vec{a}\times\vec{c}|\) | (II) | 1 |
| (C) | \(|\vec{a}+\vec{b}+\vec{c}|\) | (III) | \(\sqrt{5}\) |
| (D) | \((\vec{b}+\vec{c})\cdot(\vec{b}+\vec{a})\) | (IV) | 4 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
Since the vectors are mutually perpendicular, \(\vec{a}\cdot\vec{b} = \vec{b}\cdot\vec{c} = \vec{c}\cdot\vec{a} = 0\), and the angle between any two of them is \(\displaystyle\frac{\pi}{2}\).
(A) \(|\vec{a}+\vec{b}|\)
This matches (III) \(\sqrt{5}\).
\(|\vec{a}+\vec{b}|^2 = 4+1+0 = 5\)
(B) \(|\vec{a}\times\vec{c}|\)
This matches (IV) 4.
\( \displaystyle |\vec{a}\times\vec{c}| = 2\times2\times\sin\frac{\pi}{2} = 4 \)
(C) \(|\vec{a}+\vec{b}+\vec{c}|\)
This matches (I) 3.
\(|\vec{a}+\vec{b}+\vec{c}|^2 = 4+1+4+0 = 9\)
(D) \((\vec{b}+\vec{c})\cdot(\vec{b}+\vec{a})\)
This matches (II) 1.
\( = |\vec{b}|^2+\vec{b}\cdot\vec{a}+\vec{c}\cdot\vec{b}+\vec{c}\cdot\vec{a}\)
\( = 1+0+0+0 = 1\)