A) \[\frac{\widehat{\mathbf{i}}-\widehat{\mathbf{j}}-\widehat{\mathbf{k}}}{\sqrt{3}}\]
B) \[\frac{\widehat{\mathbf{i}}+\widehat{\mathbf{j}}+\widehat{\mathbf{k}}}{\sqrt{3}}\]
C) \[\frac{\widehat{\mathbf{i}}+\widehat{\mathbf{j}}-\widehat{\mathbf{k}}}{\sqrt{3}}\]
D) None of these
Correct Answer: A
Solution :
Let\[\overset{\to }{\mathop{\mathbf{a}}}\,=3\widehat{\mathbf{i}}+\widehat{\mathbf{j}}+2\widehat{\mathbf{k}}\]and\[\overset{\to }{\mathop{\mathbf{b}}}\,=2\widehat{\mathbf{i}}-2\widehat{\mathbf{j}}+4\widehat{\mathbf{k}}\] Now, \[\overset{\to }{\mathop{\mathbf{a}}}\,\times \overset{\to }{\mathop{\mathbf{b}}}\,=\left| \begin{matrix} \widehat{\mathbf{i}} & \widehat{\mathbf{j}} & \widehat{\mathbf{k}} \\ 3 & 1 & 2 \\ 2 & -2 & 4 \\ \end{matrix} \right|\] \[=\widehat{\mathbf{i}}(4+4)-\widehat{\mathbf{j}}(12-4)+\widehat{\mathbf{k}}(-6-2)\] \[=8\widehat{\mathbf{i}}-8\widehat{\mathbf{j}}-8\widehat{\mathbf{k}}\] \[\therefore \]Unit vector\[=\frac{\overset{\to }{\mathop{\mathbf{a}}}\,\times \overset{\to }{\mathop{\mathbf{b}}}\,}{|\overset{\to }{\mathop{\mathbf{a}}}\,\times \overset{\to }{\mathop{\mathbf{b}}}\,|}\] \[=\frac{8(\widehat{\mathbf{i}}-\widehat{\mathbf{j}}-\widehat{\mathbf{k}})}{8\sqrt{{{1}^{2}}+{{1}^{2}}+{{1}^{2}}}}\] \[=\frac{\widehat{\mathbf{i}}-\widehat{\mathbf{j}}-\widehat{\mathbf{k}}}{\sqrt{3}}\]
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