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  4. The vectors a=3i-2j+2k and b=-i-2k are the adjacent sides of a parallelogram. Then, angle between its diagonals is

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Q. The vectors $ a=3i-2j+2k $ and $ b=-i-2k $ are the adjacent sides of a parallelogram. Then, angle between its diagonals is

JamiaJamia 2013

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} $