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Question

If ¯¯¯¯¯¯¯¯AB=(3,2,2),¯¯¯¯¯¯¯¯BC=(1,02) are the adjacent sides of a parallelogram, then the obtuse angle between its diagonals is

A
π4
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B
π2
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C
π3
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D
3π4
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Solution

The correct option is D 3π4

AB=(3,2,2)=3ˆi2ˆj+2ˆk
BC=(1,0,2)=ˆi2ˆk
AC=AB+BC=(3ˆi2ˆj+2ˆk)+(ˆi2ˆk)=2ˆi2ˆj
BD=BC+BA=BC+(AB)=(ˆi2ˆk)+(3ˆi+2ˆj2ˆk)=4ˆi+2ˆj2ˆk
AC.BD=AC.BD.cosθ
(2)(4)+(2)(2)+(0)(4)=(2)2+(2)2×(4)2+(2)2+(4)2.cosθ
(8)+(4)+0=22×6×cosθ
12=22×6×cosθ
cosθ=12=cos(ππ4)=cos(3π4)
θ=3π4

1291855_1114435_ans_1a63c6b27f9440b89cc5fdb1e2c59a02.png

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