Match List-I with List-II
| List-I | List-II | ||
| (A) | \( (\vec{a}-\vec{b})\times(\vec{a}+\vec{b}) \) equals | (I) | \( \vec{0} \) |
| (B) | \( \vec{a}=\left(3\hat{\imath}+\hat{\jmath}-4\hat{k}\right),\vec{b}=\left(6\hat{\imath}+5\hat{\jmath}-2\hat{k}\right) \), then \( |\vec{a}\times\vec{b}| \) equals | (II) | 5 |
| (C) | \( \vec{a}\times(\vec{b}+\vec{c})+\vec{b}\times(\vec{c}+\vec{a})+\vec{c}\times(\vec{a}+\vec{b})= \) | (III) | \( 2(\vec{a}\times\vec{b}) \) |
| (D) | Projection of vector \( \left(\hat{\imath}+3\hat{\jmath}+7\hat{k}\right) \) on vector \( \left(2\hat{\imath}-3\hat{\jmath}+6\hat{k}\right) \) is | (IV) | 27 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) - (III), (B) - (IV), (C) - (I), (D) - (II)
Two properties of the cross product are used: \( \vec{a}\times\vec{a}=\vec{0} \) and \( \vec{b}\times\vec{a}=-\vec{a}\times\vec{b} \).
(A) \( (\vec{a}-\vec{b})\times(\vec{a}+\vec{b}) \)
This matches (III) \( 2(\vec{a}\times\vec{b}) \).
\( \displaystyle (\vec{a}-\vec{b})\times(\vec{a}+\vec{b})=\vec{a}\times\vec{a}+\vec{a}\times\vec{b}-\vec{b}\times\vec{a}-\vec{b}\times\vec{b} \)
\( \displaystyle =\vec{0}+\vec{a}\times\vec{b}+\vec{a}\times\vec{b}-\vec{0} \)
\( \displaystyle =2(\vec{a}\times\vec{b}) \)
(B) \( |\vec{a}\times\vec{b}| \) for \( \vec{a}=3\hat{\imath}+\hat{\jmath}-4\hat{k} \), \( \vec{b}=6\hat{\imath}+5\hat{\jmath}-2\hat{k} \)
This matches (IV) 27.
\( \displaystyle \vec{a}\times\vec{b}=\begin{vmatrix}\hat{\imath} & \hat{\jmath} & \hat{k}\\3 & 1 & -4\\6 & 5 & -2\end{vmatrix} \)
\( \displaystyle =\hat{\imath}(-2+20)-\hat{\jmath}(-6+24)+\hat{k}(15-6) \)
\( \displaystyle =18\hat{\imath}-18\hat{\jmath}+9\hat{k} \)
\( \displaystyle |\vec{a}\times\vec{b}|=\sqrt{324+324+81} \)
\( \displaystyle =\sqrt{729}=27 \)
(C) \( \vec{a}\times(\vec{b}+\vec{c})+\vec{b}\times(\vec{c}+\vec{a})+\vec{c}\times(\vec{a}+\vec{b}) \)
This matches (I) \( \vec{0} \). Expanding gives six terms, which pair off:
\( \displaystyle (\vec{a}\times\vec{b}+\vec{b}\times\vec{a})+(\vec{a}\times\vec{c}+\vec{c}\times\vec{a})+(\vec{b}\times\vec{c}+\vec{c}\times\vec{b})=\vec{0} \)
(D) Projection of \( \hat{\imath}+3\hat{\jmath}+7\hat{k} \) on \( 2\hat{\imath}-3\hat{\jmath}+6\hat{k} \)
This matches (II) 5. The projection of \( \vec{p} \) on \( \vec{q} \) is \( \frac{\vec{p}\cdot\vec{q}}{|\vec{q}|} \):
\( \displaystyle \frac{(1)(2)+(3)(-3)+(7)(6)}{\sqrt{4+9+36}}=\frac{2-9+42}{7} \)
\( \displaystyle =\frac{35}{7} \)
\( \displaystyle =5 \)
So the matching is (A)-(III), (B)-(IV), (C)-(I), (D)-(II).