Question

If four points A (a), B(b), C(c), and D(d) are coplanar then show that ( a b d )+( b c d )+( c a d )=( a b c )

Answer 1

Please see the attachment

Answer 2

We have to show that

( a b d) + (b c d) + (c a d) = (a b c)

• Given,

four points A(a) , B(a) , C(c) , D(d) are coplanar

therefore,

the vectors AB , AC , AD are also coplanar

• As these vectors are coplanar their scalar triple product will be 0

Therefore,

AB•(AC × AD) = 0

(b - a) • ((c - a) × (d - a)) = 0

(b - a)•((c × d) - (c × a) - (a × d) + (a × a)) = 0

As a × a = 0

hence, (b - a)•((c × d) - (c × a) - (a × d)) = 0

b•(c × d) - b•(c × a) - b•(a × d) - a•(c × d)+a•(c × a)+a•(a × d)=0

(b c d) - (b c a) - (b a d) - (a c d) + (a c a) + (a a d) = 0

• Now if any two vectors are same then the scalar product is zero therefore,

(a c a) = 0 and (a a d) = 0

we have,

(b c d) - (b c a) - (b a d) - (a c d) = 0 - (1)

• Now,

(b c a) = (a b c)

-(b a d) = (a b d)

-(a c d) = (c a d)

therefore, (1) can be written as

(a b d) + (b c d) + (c a d) = (a b c), hence proved.