Question

Prove that the four point A, B, C and D are coplanar if and only if AD.(AB×AC)=0




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Answer 1

Answer:

ABCD are coplanar if and only if AD~ = kAB~ + lAC~ for some k, l

if and only if d − a = k(b − a) + l(c − a)

if and only if a(k + l − 1) + b(−k) + c(−l) + d = 0

⇒ αa + βb + γc + δd = 0 and α + β + γ + δ = 0

Conversely if αa + βb + γc + δd = 0 and α + β + γ + δ = 0

If δ = 0

αa + βb + γc = 0 and α + β + γ = 0 So A, B, C are colinear

Thus ABCD are coplanar.

If δ 6= 0

α

δ

a +

β δ b + γ δ

c + d = 0

Let β

δ = −k

γ

δ = −l then α

δ = k + l − 1

Hence ABCD are coplanar.