A) \[2\hat{i}+\hat{j}+5\hat{k}\]
B) \[2\hat{i}+3\hat{j}-3\hat{k}\]
C) \[2\hat{i}-\hat{j}+5\hat{k}\]
D) \[2\hat{i}+3\hat{j}+3\hat{k}\]
Correct Answer: B
Solution :
Given, \[a=2\hat{i}-\hat{j}+\hat{k},\,b=\hat{i}+2\hat{j}-\hat{k}\] and \[c=\hat{i}+\hat{j}-2\hat{k}\] Now, we have; \[b+\lambda c=(1+\lambda )\hat{i}+(2+\lambda )\hat{j}+(-1-2\lambda )\hat{k}\] \[\therefore \] Projection of \[(b+\lambda c)\] on \[a=\left| \frac{(b+\lambda c)\cdot a}{|a|} \right|=\sqrt{\frac{2}{3}}\] (given) \[\Rightarrow \] \[\left| \frac{2(1-\lambda )-(2+\lambda )\,+(-1-2\lambda )}{\sqrt{4+1+1}} \right|=\sqrt{\frac{2}{3}}\] \[\Rightarrow \] \[\left| \frac{-\lambda -1}{\sqrt{6}} \right|=\frac{\sqrt{2}}{\sqrt{3}}\Rightarrow \,\,\lambda +1=2\Rightarrow \,\,\lambda =1\] \[\therefore \] \[b+\lambda c=2\hat{i}+3\hat{j}-3\hat{k}\]You need to login to perform this action.
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