A) \[\frac{-4\hat{i}+3\hat{j}-\hat{k}}{\sqrt{26}}\]
B) \[\frac{-4\hat{i}+3\hat{j}+\hat{k}}{\sqrt{26}}\]
C) \[\frac{-3\hat{i}+2\hat{j}-\hat{k}}{\sqrt{14}}\]
D) \[\frac{-3\hat{i}+2\hat{j}+\hat{k}}{\sqrt{14}}\]
Correct Answer: B
Solution :
[b] \[\vec{A}=\hat{i}+\hat{j}+\hat{k}\] \[\vec{B}=2\hat{i}+3\hat{j}-\hat{k}\] \[\vec{A}\times \vec{B}=\left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 1 & 1 & 1 \\ 2 & 3 & -1 \\ \end{matrix} \right|\] \[=\hat{i}(-1-3)-\hat{j}(-1-2)+\hat{k}(3-2)\] \[=-4\hat{i}+3\hat{j}+\hat{k}\] Vector of unit length orthogonal to both the vectors \[\overset{\to }{\mathop{A}}\,\] and \[\overset{\to }{\mathop{B}}\,\] \[=\frac{\overset{\to }{\mathop{A}}\,\times \overset{\to }{\mathop{B}}\,}{|\overset{\to }{\mathop{A}}\,\times \overset{\to }{\mathop{B}}\,|}\] \[=\frac{-\,4i+3j+k}{\sqrt{16+9+1}}=\frac{-\,4\hat{i}+3\hat{j}+\hat{k}}{\sqrt{26}}\]
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