A) \[\frac{-\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}\]
B) \[\frac{\hat{i}+\hat{j}-\hat{k}}{3}\]
C) \[\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}\]
D) \[\frac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}\]
Correct Answer: D
Solution :
Let \[\vec{a}=\hat{i}+\hat{j}\] and \[\vec{b}=\hat{j}+\hat{k}\] Now, \[\vec{a}\times \vec{b}=\left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 1 & 1 & 0 \\ 0 & 1 & 1 \\ \end{matrix} \right|\] \[=\hat{i}(1-0)-\hat{j}(1-0)+\hat{k}(1-0)\]\[=\hat{i}-\hat{j}+\hat{k}\] and \[|\vec{a}\times \vec{b}|=\sqrt{{{1}^{2}}+{{(-1)}^{2}}+{{1}^{2}}}=\sqrt{3}\] \[\therefore \]Required unit vector \[=\frac{\vec{a}\times \vec{b}}{|\vec{a}\times \vec{b}|}\]\[=\frac{\hat{i}-\hat{j}+\hat{k}}{\sqrt{3}}\]
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