If A = 3i-2j-k and B=2i+4j+2k, find...
Question

# If ( overrightarrow{mathrm{A}}=3 hat{mathrm{i}}-2 hat{mathrm{j}}-hat{mathrm{k}} ) and ( overrightarrow{mathrm{B}}=2 hat{mathrm{i}}+4 hat{mathrm{j}}+2 hat{mathrm{k}}, ) find ( |overrightarrow{mathrm{A}}|,|overrightarrow{mathrm{B}}| ) and ( |overrightarrow{mathrm{A}}+overrightarrow{mathrm{B}}| cdot ) Also find the direction of ( overrightarrow{mathrm{A}}+overrightarrow{mathrm{B}} )with the ( x ) -axis. Check whether ( |overrightarrow{mathrm{A}}|+|overrightarrow{mathrm{B}}| ) is equal to ( |overrightarrow{mathrm{A}}+overrightarrow{mathrm{B}}| )

JEE/Engineering Exams
Physics
Solution
92
4.0 (1 ratings)

[
begin{array}{l}
vec{A}=3 hat{i}-2 hat{jmath}-hat{k} ; vec{B}=2 hat{imath}+4 hat{jmath}+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{imath}-2 hat{imath}-hat{k}+2 hat{imath}+y hat{jmath}+2 hat{k}
end{array}
]
(5hat{imath}+12 hat{j}+hat{k} )
( |vec{A}+vec{B}|=sqrt{5^{2}+2^{2}+1^{2}}=sqrt{25+4+1}=sqrt{30} )
Unit vector along ( vec{x}-operatorname{aris}=hat{imath} )
( (vec{A}+vec{B}) cdot hat{imath}=|| vec{A}+vec{B}|| hat{x} 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 ^{-1} sqrt{frac{5}{6}} )
[
|vec{A}|+|vec{B}|=sqrt{14}+sqrt{24}
]
Uleasly ( |vec{A}|+|vec{B}| eq|vec{A}+vec{B}| )