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a, c unit vectors at 60 degrees with (a x (b x c)).(a x c) = 5: value of [a b c]

The same question, asked 2 times: 10 May 2023 S1*, 21 Apr 2025 S1. Every wording below, oldest first.

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  1. Question 1

    10 May 2023 S1

    121
    If a⃗\vec{a} and c⃗\vec{c} are unit vectors inclined at π3\frac{\pi}{3} with each other and (a⃗×(b⃗×c⃗))⋅(a⃗×c⃗)=5(\vec{a}\times(\vec{b}\times\vec{c}))\cdot(\vec{a}\times\vec{c})=5, then the value of 5[a⃗ b⃗ c⃗]=5[\vec{a}\,\vec{b}\,\vec{c}]=

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  2. Question 2

    21 Apr 2025 S1

    110
    Let aˉ\bar{a} and cˉ\bar{c} be unit vectors at an angle π3\frac{\pi}{3} with each other. If (aˉ×(bˉ×cˉ))⋅(aˉ×cˉ)=5(\bar{a}\times (\bar{b}\times\bar{c})) \cdot (\bar{a}\times\bar{c}) = 5, then [a‾b‾c‾]=\lbrack\overline{a}\overline{b}\overline{c}\rbrack =

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