If ((â × b̂) × (ĉ × d̂)) · (â × d̂) = 0, which statement must always hold?

The vectors â, b̂, ĉ, d̂ all lie in one plane.
Either â or d̂ lies in the plane formed by b̂ and ĉ.
Either b̂ or ĉ lies in the plane formed by â and d̂.
Either b̂ or d̂ lies in the plane formed by â and ĉ.

Correct answer: Either b̂ or ĉ lies in the plane formed by â and d̂.

Solution

(a×b)×(c×d) = [a b d]c - [a b c]d. Dotting with (a×d) and using d.(a×d)=0 gives [a b d][a d c] = 0. Hence either [a b d]=0 (b in plane of a,d) or [a d c]=0 (c in plane of a,d): either b or c lies in the plane of a and d.

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