Let \(\vec{a}=2\hat{\imath}+\hat{\jmath}-2\hat{k}\) and \(\vec{b}=3\hat{\jmath}+4\hat{k}\).
Match List-I with List-II
| List-I | List-II | ||
| (A) | \(\vec{a}\cdot\vec{b}\) | (I) | −1 |
| (B) | \(|\vec{a}|\) | (II) | 3 |
| (C) | Projection of \(\vec{a}\) on \(\vec{b}\) | (III) | \(10\sqrt{2}\) |
| (D) | \(|\vec{a}\times\vec{b}|\) | (IV) | −5 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(IV), (B)-(II), (C)-(I), (D)-(III)
(A) \(\vec{a}\cdot\vec{b}\)
This matches (IV) −5.
\(\vec{a}\cdot\vec{b} = (2)(0)+(1)(3)+(-2)(4) = -5\)
(B) \(|\vec{a}|\)
This matches (II) 3.
\(|\vec{a}| = \sqrt{4+1+4} = 3\)
(C) Projection of \(\vec{a}\) on \(\vec{b}\)
This matches (I) −1.
Here \(|\vec{b}| = \sqrt{9+16} = 5\), so the projection is
\( \displaystyle \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|} = \frac{-5}{5} = -1 \)
(D) \(|\vec{a}\times\vec{b}|\)
This matches (III) \(10\sqrt{2}\).
\( \displaystyle \vec{a}\times\vec{b} = \begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ 2 & 1 & -2 \\ 0 & 3 & 4 \end{vmatrix} \)
\( = \hat{\imath}(4+6)-\hat{\jmath}(8-0)+\hat{k}(6-0)\)
\( = 10\hat{\imath}-8\hat{\jmath}+6\hat{k}\)
\(|\vec{a}\times\vec{b}| = \sqrt{100+64+36} = \sqrt{200} = 10\sqrt{2}\)