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Question

The vectors $$\overline {AB} = \overline {3i} - \overline {2j} + \overline {2k}$$ and $$\overline {BC} = -\overline {i} - \overline {2k}$$ are the adjacent sides of a parallelogram. The angle between its diagonals is


A
π2
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B
π3 or 2π3
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C
3π4 or π4
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D
5π6 or π6
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Solution

The correct option is B $$\dfrac {3\pi}{4}$$ or $$\dfrac {\pi}{4}$$
let diagonals be AC and BD

$$\vec { AC } =\vec { AB } +\vec { BC } =2i-2j$$

$$\vec { BD } =\vec { BC } +\vec { CD } =\vec { BC } -\vec { AB } =-4i+2j-4k$$
$$\therefore \cos { \varphi  } =\dfrac { \vec { AC } \cdot \vec { BD }  }{ \left| \vec { AC }  \right| \left| \vec { BD }  \right|  } =\dfrac { -8-4 }{ 2\sqrt { 2 } \cdot 6 } $$

$$\therefore \cos { \varphi  } =\dfrac { -1 }{ \sqrt { 2 }  } $$

$$\therefore \varphi =\dfrac { \pi  }{ 4 } $$ or $$\dfrac { 3\pi  }{ 4 } $$

Mathematics

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