A) \[\frac{\hat{i}+\hat{j}}{\sqrt{2}}\]
B) \[\hat{k}\]
C) \[\frac{\hat{j}+\hat{k}}{\sqrt{2}}\]
D) \[\frac{\hat{i}-\hat{j}}{\sqrt{2}}\]
Correct Answer: A
Solution :
[a] According to question \[a=-\hat{i}+\hat{j}+\hat{k}\] and \[b=\hat{i}-\hat{j}+\hat{k}\] Then, \[a\times b=\left| \begin{matrix} i & j & k \\ -1 & 1 & 1 \\ 1 & -1 & 1 \\ \end{matrix} \right|\] \[=\hat{i}[1+1]-\hat{j}[-1-1]+\hat{k}[1-1]\] \[=2\hat{i}+2j+0=2(i+j)\] and \[|a\times b|=\sqrt{4+4}=2\sqrt{2}\] \[\therefore \] Required unit vector \[=\pm \frac{2(i+j)}{2\sqrt{2}}=\pm \frac{i+j}{\sqrt{2}}\]
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