A) \[\pm \frac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{6}}\]
B) \[\frac{\hat{i}+\hat{j}-2\hat{k}}{\sqrt{6}}\]
C) \[\pm \frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{6}}\]
D) \[\frac{\hat{i}-\hat{j}+2\hat{k}}{\sqrt{6}}\]
Correct Answer: B
Solution :
\[\vec{d}=x\hat{i}+y\hat{j}+z\hat{k}\] \[\therefore \,\,\vec{a}.\vec{d}=0\] gives \[x-y=0\] ...(i) And \[[\vec{b}\,\vec{c}\,\vec{d}]\,=0\] gives \[x+y+z=0\] ?(ii) \[\therefore \,\,x=y\] and \[z=-2x\] And \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}=1\] \[\therefore \,\,x=\pm \,\frac{1}{\sqrt{6}};\,\,y=\pm \,\frac{1}{\sqrt{6}},\,z=\mp \,\frac{2}{\sqrt{6}}\] \[\therefore \,\,d=\pm \,\frac{\hat{i}+\hat{j}-2\hat{k}}{\sqrt{6}}\]
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