If A=3i^−2j^−k^ and B=2i^+4j +2k^,...
Question

If ( bar{A}=3 hat{i}-2 hat{j}-hat{k} ) and ( bar{B}=2 hat{i}+4 vec{j}+2 hat{k}, ) find ( |bar{A}|,|bar{B}| ) and ( |bar{A}+bar{B}| ). Also find the diection of ( bar{A}+bar{B} )with the x-axais. Check whether ( |vec{A}|+|vec{B}| ) is equal to ( mid vec{A}+vec{B} )

JEE/Engineering Exams
Physics
Solution
102
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( vec{A}=3 hat{i}-2 hat{j}-hat{k} ; vec{B}=2 hat{i}+4 hat{j}+2 hat{k} )
( |vec{A}|=sqrt{3^{2}+2^{2}+1^{2}}=sqrt{9+4+1}=sqrt{14} )
[
|vec{B}|=sqrt{2^{2}+4^{2}+2^{2}}=sqrt{4+16+4}=sqrt{24}
]
( vec{A}+vec{B}=3 hat{i}-2 hat{imath}-hat{k}+2 hat{i}+4 hat{j}+2 hat{k} )
( =5 hat{i}+2 hat{j}+hat{k} )
( |vec{A}+vec{B}|=sqrt{5^{2}+2^{2}+1^{2}}=sqrt{25+4+1}=sqrt{30} )
Unit veetor along ( vec{x}-operatorname{arcs}=hat{imath} )
( (vec{A}+vec{B}) cdot hat{I}=|| vec{A}+vec{B}|| hat{lambda} mid cdot cos theta )
( Rightarrow(5 hat{imath}+2 hat{jmath}+hat{k}) cdot hat{imath}=sqrt{30} cos theta )
( Rightarrow quad 5=sqrt{30} cos theta )
( Rightarrow cos theta=frac{5}{sqrt{30}}=sqrt{frac{5}{6}} )
( theta=cos ^{4} sqrt{frac{5}{6}} )
( |vec{A}|+|vec{B}|=sqrt{14}+sqrt{24} )
Learly ( |vec{A}|+|vec{B}| eq|vec{A}+vec{B}| )