A) \[\frac{\pi }{4}\]
B) \[\frac{\pi }{3}\]
C) \[\frac{\pi }{2}\]
D) \[\frac{2\pi }{3}\]
Correct Answer: A
Solution :
The diagonals of the parallelogram are \[\overrightarrow{\alpha }=\overrightarrow{a}+\overrightarrow{b}\]and\[\overrightarrow{\beta }=\overrightarrow{a}-\overrightarrow{b}\] i.e.,\[\overrightarrow{\alpha }=2i-2j\]and \[\overrightarrow{\beta }=\pm (4i-2j+4k)\] Let \[\theta \]be the angle between the diagonals Then, \[\cos \theta =\frac{\overrightarrow{\alpha }.\overrightarrow{\beta }}{|\overrightarrow{\alpha }|.|\overrightarrow{\beta }|}\] \[\Rightarrow \] \[\cos \theta =\frac{1}{\sqrt{2}}\]or\[\cos \theta =-\frac{1}{\sqrt{2}}\] \[\Rightarrow \] \[\theta =\frac{\pi }{4}\]or\[\theta =\frac{3\pi }{4}\]
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