A) \[\hat{j}-\hat{k}\]
B) \[\frac{\hat{i}+\hat{j}}{\sqrt{2}}\]
C) \[\frac{\hat{j}+\hat{k}}{\sqrt{2}}\]
D) \[\frac{\hat{j}-\hat{k}}{\sqrt{2}}\]
E) \[5(\hat{j}-\hat{k})\]
Correct Answer: D
Solution :
Let vector coplanar with\[\hat{i}+2\hat{j}+\hat{k}\]and \[\hat{i}+\hat{j}+2\hat{k}\]is \[x(\hat{i}+2\hat{j}+\hat{k})+y(\hat{i}+\hat{j}+2\hat{k})\] \[=(x+y)\hat{i}+(2x+y)\hat{j}+(x+2y)\hat{k}\] This vector is perpendicular to vector \[(2\hat{i}+\hat{j}+\hat{k})\]. \[\therefore \]\[[(x+y)\hat{i}+(2x+y)\hat{j}+(x+2y)\hat{k}]\] \[.(2\hat{i}+\hat{j}+\hat{k})=0\] \[\Rightarrow \] \[2x+2y+2x+y+x+2y=0\] \[\Rightarrow \] \[5x+5y=0\] \[\Rightarrow \] \[y=-x\] \[\therefore \]Equation of vector coplanar with given vectors is\[0\hat{i}+x\hat{j}-x\hat{k}\]. \[\therefore \]Required unit vector is \[\frac{x(\hat{j}-\hat{k})}{x\sqrt{1+1}}=\frac{\hat{j}-\hat{k}}{\sqrt{2}}\]
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