A,B,C,D are four points in the space and satisfy ∣vec(AB)∣=3,∣vec(BC)∣=7,∣vec(CD)∣=11 and ∣vec(DA)∣=9. Then vec(AC).vec(BD) is I know that vec(AB)+vec(BC)+vec(CA)+vec(AD)=0 and also I observed that ∣vec(AB)∣^2+∣vec(CD)∣^2=∣vec(BC)∣^2+∣vec(DA)∣^2. But couldn't get anything from it. Any hint?

2k1ablakrh0 2022-09-27 Answered
A,B,C,D are four points in the space and satisfy A B ∣= 3 , B C ∣= 7 , C D ∣= 11 and D A ∣= 9. Then A C . B D is
I know that A B + B C + C A + A D = 0 and also I observed that
A B 2 + C D 2 =∣ B C 2 + D A 2 . But couldn't get anything from it. Any hint?
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Answers (1)

pedradauy
Answered 2022-09-28 Author has 8 answers
A C B D = ( D A + D C ) ( D B ) = D A D B D C D B =
= D A D B cos A D B D C D B cos C D B =
= 9 2 + D B 2 3 2 2 11 2 + D B 2 7 2 2 = 0.
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