A) \[\frac{\pi }{3}\]
B) \[\frac{\pi }{2}\]
C) \[\frac{\pi }{4}\]
D) \[\frac{2\pi }{3}\]
Correct Answer: D
Solution :
Let \[\overset{\to }{\mathop{\mathbf{a}}}\,\] and \[\overset{\to }{\mathop{\mathbf{b}}}\,\] be two unit vectors \[\therefore \] \[|\overset{\to }{\mathop{\mathbf{a}}}\,|\,\,=\,\,|\overset{\to }{\mathop{\mathbf{b}}}\,|\,\,=1\] Now, \[|\overset{\to }{\mathop{\mathbf{a}}}\,+\overset{\to }{\mathop{\mathbf{b}}}\,{{|}^{2}}=1\] (given) \[|\overset{\to }{\mathop{\mathbf{a}}}\,{{|}^{2}}+|\overset{\to }{\mathop{\mathbf{b}}}\,{{|}^{2}}+2\overset{\to }{\mathop{\mathbf{a}}}\,\cdot \overset{\to }{\mathop{\mathbf{b}}}\,=1\] \[2\overset{\to }{\mathop{\mathbf{a}}}\,\cdot \overset{\to }{\mathop{\mathbf{b}}}\,=1-1-1=-1\] \[\overset{\to }{\mathop{\mathbf{a}}}\,\cdot \overset{\to }{\mathop{\mathbf{b}}}\,=\frac{-1}{2}\] \[\therefore \] \[|\overset{\to }{\mathop{\mathbf{a}}}\,|\cdot |\overset{\to }{\mathop{\mathbf{b}}}\,|cos\theta =-\frac{1}{2}\] \[\Rightarrow \] \[\theta =\frac{2\pi }{3}\]
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