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
B
π3 or 2π3
C
3π4 or π4
D
5π6 or π6

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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