A, B, C, D are four points in space. using vector methods, prove that
AC 2 + BD 2 + AD 2 + BC 2 ≥ AB 2 + CD 2 what is the implication of the sign of equality
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
Let the position vector of A, B, C, D be 
and 
respectively then
AC 2 +BD 2 + AD 2 + BC 2 = 
. 
+ 
. 
+ 
. 
+ 
. 
= 
+ 
–2 
+ 
+ 
– 2 
+ 
+ 
– 2 
+ 
+ 
– 
= 
+ 
– 2 
+ 
+ 
– 2 
+ 
+ 
+ 
+ 
+ 2 
+ 
– 
– 
– 
– 
= 
. 
+ 
. 
+ 
= AB 2 + CD 2 + 
. 
≥ AB 2 + CD 2
⇒ AC 2 + BD 2 + AD 2 + BC 2 ≥ AB 2 + CD 2
for the sign of equality to hold, 
= 0 or 
⇒ 
and 
are collinear, the four points A, B, C, D are collinear
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