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Q. Given two vectors; $\vec{ A }=\hat{ i }+\hat{ j }$ and $\vec{ B }=\hat{ i }-\hat{ j }$. Then match the following columns :

Column I Column II

A $(\vec{ A }+\vec{ B }) / 2$ 1 $\hat{ i }$

B $(\vec{ A }-\vec{ B }) / 2$ 2 $\hat{ j }$

C $(\vec{ A } \cdot \vec{ B }) / 2$ 3 $-\hat{ k }$

D $(\vec{ A } \times \vec{ B }) / 2$ 4 $0$

Motion in a Plane

A

( A )$\rightarrow$(4) ; (B) $\rightarrow$ (1); C $\rightarrow$ (2); (D) $\rightarrow$ (2)

5%

B

(A) $\rightarrow$ (2); (B) $\rightarrow$ (4); C$ \rightarrow$ (3); (D) $\rightarrow$ (1)

8%

C

(A) $\rightarrow$ (3); (B) $\rightarrow$ (2); C $\rightarrow$ (4); (D) $\rightarrow$ (1)

14%

D

(A) $\rightarrow$ (1); (B) $\rightarrow$ (2) ; C $\rightarrow$ (4); (D) $\rightarrow$ (3)

73%

Solution:

$A \rightarrow(1) ; B \rightarrow(2) ; C \rightarrow(4) ; D \rightarrow(3)$
(A) $(\vec{A}+\vec{B}) / 2=\frac{(\hat{i}+\hat{j})+(\hat{i}-\hat{j})}{2}=\hat{i}$
(B) $(\vec{A}-\vec{B}) / 2=\frac{(\hat{i}+\hat{j})-(\hat{i}-\hat{j})}{2}=\hat{j}$
(C) $(\vec{A} \cdot \vec{B}) / 2=\frac{(\hat{i}+\hat{j}) \cdot(\hat{i}-\hat{j})}{2}=\frac{1-1}{2}=0$
(D) $\frac{(\vec{A} \times \vec{B})}{2}=\frac{(\hat{i}+\hat{j}) \times(\hat{i}-\hat{j})}{2}=\frac{0-\hat{k}-\hat{k}+0}{2}=-\hat{k}$