Question

if a b c d are coplanar then prove (a×b)×(c×d)=0

Answer 1

Your answer mate

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I assumed the four points A, B, C and D with position vector a, b, c and d to be on the x-y plane. So cross product between any of them is towards the z-axis. So I expanded the equation :

[a^b^d^]+[b^c^d^]+[c^a^d^]=[a^b^c^][a^b^d^]+[b^c^d^]+[c^a^d^]=[a^b^c^]

a^⋅(b^×c^)+b^⋅(c^×d^)+c^⋅(a^×d^)=a^⋅(b^×c^)a^⋅(b^×c^)+b^⋅(c^×d^)+c^⋅(a^×d^)=a^⋅(b^×c^)

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