A) \[\pi \]
B) \[\frac{2\pi }{3}\]
C) \[\frac{\pi }{4}\]
D) \[\frac{\pi }{2}\]
Correct Answer: C
Solution :
Given, \[|\overset{\to }{\mathop{\mathbf{a}}}\,\times \overset{\to }{\mathop{\mathbf{b}}}\,|=|\overset{\to }{\mathop{\mathbf{a}}}\,\cdot \overset{\to }{\mathop{\mathbf{b}}}\,|\] \[\therefore \] \[ab\sin \theta |\widehat{\mathbf{n}}|=ab\cos \theta \] \[\Rightarrow \] \[\tan \theta =1\] \[(\because \,\,|\widehat{\mathbf{n}}|=1)\] \[\Rightarrow \] \[\theta =\frac{\pi }{4}\] Note: If \[\overset{\to }{\mathop{\mathbf{a}}}\,\,\,\text{and}\,\,\overset{\to }{\mathop{\mathbf{b}}}\,\] are two vectors, then \[\overset{\to }{\mathop{\mathbf{a}}}\,\times \overset{\to }{\mathop{\mathbf{b}}}\,=|\overset{\to }{\mathop{\mathbf{a}}}\,||\overset{\to }{\mathop{\mathbf{b}}}\,|\]\[\sin \,\,\,\theta \,\,\,\widehat{\mathbf{n}}\] where \[\widehat{\mathbf{n}}\] is a unit vector which is perpendicular to the vectors\[\overset{\to }{\mathop{\mathbf{a}}}\,\]and\[\overset{\to }{\mathop{\mathbf{b}}}\,\].
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