A) \[{{90}^{o}}\]
B) \[{{45}^{o}}\]
C) \[{{30}^{o}}\]
D) \[{{15}^{o}}\]
Correct Answer: A
Solution :
Given, \[\vec{a}=(1,1,4)=\hat{i}+\hat{j}+4\hat{k}\] and \[\vec{b}=(1,-1,4)=\hat{i}-\hat{j}+4\hat{k}\] \[\therefore \] \[\vec{a}+\vec{b}=2\hat{i}+8\hat{k}\] and \[\vec{a}-\vec{b}=2\hat{j}\] Let \[\theta \] be the angle between \[\vec{a}+\vec{b}\]and \[\vec{a}-\vec{b},\] then \[\cos \theta =\frac{(\vec{a}+\vec{b}).(\vec{a}-\vec{b})}{|\vec{a}+\vec{b}||\vec{a}-\vec{b}|}\] \[=\frac{(2\hat{i}+0\hat{j}+8\hat{k}).(0\hat{i}+2\hat{j}+0\hat{k})}{\sqrt{{{2}^{2}}+{{0}^{2}}+{{8}^{2}}}\,\sqrt{{{0}^{2}}+{{2}^{2}}+{{0}^{2}}}}\] \[=0\] \[\Rightarrow \] \[\theta =\frac{\pi }{2}={{90}^{o}}\]
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