question_answer

If \[\overrightarrow{A}=4\hat{i}-3\hat{j}\] and \[\overrightarrow{B}=6\hat{i}+8\hat{j}\] then magnitude and direction of \[\overrightarrow{A}\,+\overrightarrow{B}\] will be

A)                 \[5,\,{{\tan }^{-1}}(3/4)\]            

B)                 \[5\sqrt{5},{{\tan }^{-1}}(1/2)\]

C)                 \[10,\,{{\tan }^{-1}}(5)\]              

D)                 \[25,\,{{\tan }^{-1}}(3/4)\]

Correct Answer: B

Solution :

                    \[\overrightarrow{A}+\overrightarrow{B}=4\hat{i}-3\hat{j}+6\hat{i}+8\hat{j}\,\,=10\hat{i}+5\hat{j}\]                 \[|\overrightarrow{A}+\overrightarrow{B}|\,=\,\sqrt{{{(10)}^{2}}+{{(5)}^{2}}}\] \[=\,5\sqrt{5}\]                 \[\tan \theta =\frac{5}{10}=\frac{1}{2}\] \[\Rightarrow \]\[\theta ={{\tan }^{-1}}\left( \frac{1}{2} \right)\]