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|a x (b x c)| when a is coplanar with b and c, given |a|, |b|, b.c and the angle between b and c

The same question, asked 4 times: 14 May 2024 S1*, 14 May 2024 S2*, 26 Apr 2025 S1, 26 Apr 2025 S2. Every wording below, oldest first.

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  1. Question 1

    14 May 2024 S1

    103
    Let a⃗,b⃗,c⃗\vec{a},\vec{b},\vec{c} be the vectors such that ∣a⃗∣=3,∣b⃗∣=5,b⃗⋅c⃗=10|\vec{a}|=\sqrt{3},|\vec{b}|=5,\vec{b}\cdot\vec{c}=10 and the angle between b⃗\vec{b} and c⃗\vec{c} is π3\dfrac{\pi}{3}. If a⃗\vec{a} is perpendicular to the vector b⃗×c⃗\vec{b}\times\vec{c}, then ∣a⃗×(b⃗×c⃗)∣|\vec{a}\times(\vec{b}\times\vec{c})| is equal to

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  2. Question 2

    14 May 2024 S2

    141
    If ∣a⃗∣=3,∣b⃗∣=5,b⃗⋅c⃗=10|\vec{a}|=3, |\vec{b}|=5, \vec{b}\cdot\vec{c}=10, angle between b⃗\vec{b} and c⃗\vec{c} is π3\frac{\pi}{3}, a⃗\vec{a} is perpendicular to b⃗×c⃗\vec{b}\times\vec{c}. Then the value of ∣a⃗×(b⃗×c⃗)∣|\vec{a}\times(\vec{b}\times\vec{c})| is

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  3. Question 3

    26 Apr 2025 S1

    122
    If aˉ,bˉ,cˉ\bar{a},\bar{b},\bar{c} are three coplanar vectors such that ∣a‾∣=1,∣ b‾∣=2, b‾⋅c‾=8|\overline{a}| = 1,|\overline{\text{ }b}| = 2,\overline{\text{ }b}\cdot\overline{c}= 8, the angle between b‾\overline{b} and c‾\overline{c} is 45∘45^{\circ}, then ∣a‾×(b‾×c‾)∣=|\overline{a}\times (\overline{b}\times\overline{c})| =

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  4. Question 4

    26 Apr 2025 S2

    135
    If aˉ,bˉ,cˉ\bar{a},\bar{b},\bar{c} are three coplanar vectors such that ∣a‾∣=1,∣ b‾∣=2, b‾⋅c‾=8|\overline{a}| = 1,|\overline{\text{ }b}| = 2,\overline{\text{ }b}\cdot\overline{c}= 8 and the angle between b‾\overline{b} and cˉ\bar{c} is 45∘45^{\circ} then the value of ∣aˉ×(bˉ×cˉ)∣|\bar{a}\times (\bar{b}\times\bar{c})| is

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