Let \( \vec{a} \) and \( \vec{b} \) be any two vectors such that \( \vec{a}=2\hat{\imath}+\hat{\jmath}+3\hat{k} \) and \( \vec{b}=3\hat{\imath}+5\hat{\jmath}-2\hat{k} \) then
Match the List-I with List-II
| List-I | List-II | ||
| (A) | \( |\vec{a}| \) | (I) | 5 |
| (B) | \( |\vec{b}| \) | (II) | \( \sqrt{507} \) |
| (C) | \( |\vec{a}.\vec{b}| \) | (III) | \( \sqrt{38} \) |
| (D) | \( |\vec{a}\times\vec{b}| \) | (IV) | \( \sqrt{14} \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) - (IV), (B) - (III), (C) - (I), (D) - (II)
(A) \( |\vec{a}| \)
This matches (IV) \( \sqrt{14} \).
\( \displaystyle |\vec{a}|=\sqrt{2^{2}+1^{2}+3^{2}}=\sqrt{14} \)
(B) \( |\vec{b}| \)
This matches (III) \( \sqrt{38} \).
\( \displaystyle |\vec{b}|=\sqrt{3^{2}+5^{2}+(-2)^{2}}=\sqrt{38} \)
(C) \( |\vec{a}.\vec{b}| \)
This matches (I) 5.
\( \vec{a}\cdot\vec{b}=(2)(3)+(1)(5)+(3)(-2) \)
\( =6+5-6=5 \)
(D) \( |\vec{a}\times\vec{b}| \)
This matches (II) \( \sqrt{507} \).
\( \displaystyle \vec{a}\times\vec{b}=\begin{vmatrix}\hat{\imath} & \hat{\jmath} & \hat{k}\\2 & 1 & 3\\3 & 5 & -2\end{vmatrix} \)
\( \displaystyle =(-2-15)\hat{\imath}-(-4-9)\hat{\jmath}+(10-3)\hat{k} \)
\( \displaystyle =-17\hat{\imath}+13\hat{\jmath}+7\hat{k} \)
\( \displaystyle |\vec{a}\times\vec{b}|=\sqrt{289+169+49} \)
\( \displaystyle =\sqrt{507} \)
So the matching is (A) - (IV), (B) - (III), (C) - (I), (D) - (II).
Shortcut: For (D), use Lagrange’s identity with the values of (A), (B) and (C):
\( \displaystyle |\vec{a}\times\vec{b}|^{2}=|\vec{a}|^{2}|\vec{b}|^{2}-(\vec{a}\cdot\vec{b})^{2} \)
\( \displaystyle =14\times 38-25 \)
\( \displaystyle =507 \)