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Scalar triple product of a x b, b x c, c x a in terms of [a b c]

This type of question, asked 4 times: 16 May 2023 S1, 16 Apr 2026 S2, 14 May 2026 S1, 14 May 2026 S2. All 4 questions below, oldest first.

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  1. Question 1

    16 May 2023 S1

    115
    If [a⃗×b⃗    b⃗×c⃗    c⃗×a⃗]=λ[a⃗  b⃗  c⃗]2[\vec{a}\times\vec{b}\;\; \vec{b}\times\vec{c}\;\; \vec{c}\times\vec{a}] = \lambda[\vec{a}\;\vec{b}\;\vec{c}]^2, then λ\lambda is equal to

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  2. Question 2

    16 Apr 2026 S2

    92
    The volume of a parallelopiped with coterminous edges aˉ,bˉ,cˉ\bar{a}, \bar{b}, \bar{c} is 3 cubic units. The volume (in cubic units) of a tetrahedron with coterminous edges (aˉ×bˉ),(aˉ×2cˉ),(bˉ×2cˉ)(\bar{a} \times \bar{b}), (\bar{a} \times 2\bar{c}), (\bar{b} \times 2\bar{c}) is...

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  3. Question 3

    14 May 2026 S1

    90
    If aˉ,bˉ,cˉ\bar{a}, \bar{b}, \bar{c} are vectors such that [aˉ bˉ cˉ]=3[\bar{a}\ \bar{b}\ \bar{c}] = 3, then [aˉ×bˉ  bˉ×cˉ  cˉ×aˉ]=[\bar{a} \times \bar{b}\ \ \bar{b} \times \bar{c}\ \ \bar{c} \times \bar{a}] =

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  4. Question 4

    14 May 2026 S2

    90
    If the volume of tetrahedron with its coterminous edges aˉ,bˉ,cˉ\bar{a}, \bar{b}, \bar{c} is 66 cu. units then the volume of a tetrahedron with coterminous edges (aˉ×bˉ),(bˉ×cˉ),(cˉ×aˉ)(\bar{a} \times \bar{b}), (\bar{b} \times \bar{c}), (\bar{c} \times \bar{a}) is

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