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|a| = |b| = 1, |c| = 2 and a x (a x c) + b = 0: acute angle between a and c

The same question, asked 5 times: 4 May 2023 S1, 10 May 2023 S1, 14 May 2024 S1*, 13 Apr 2026 S1, 18 Apr 2026 S2. Every wording below, oldest first.

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  1. Question 1

    4 May 2023 S1

    139
    Let a⃗,b⃗\vec{a},\vec{b} and c⃗\vec{c} be three vectors having magnitude 1, 1 and 2 respectively. If a⃗×(a⃗×c⃗)+b⃗=0⃗\vec{a}\times(\vec{a}\times\vec{c})+\vec{b}=\vec{0}, then the acute angle between a⃗\vec{a} and c⃗\vec{c} is

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  2. Question 2

    10 May 2023 S1

    113
    Let a⃗,b⃗\vec{a},\vec{b} and c⃗\vec{c} be three vectors having magnitudes 1, 1 and 2 respectively. If a⃗×(a⃗×c⃗)+b⃗=0⃗\vec{a}\times(\vec{a}\times\vec{c})+\vec{b}=\vec{0}, then the acute angle between a⃗\vec{a} and c⃗\vec{c} is

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  3. Question 3

    14 May 2024 S1

    141
    If a⃗,b⃗,c⃗\vec{a},\vec{b},\vec{c} are three vectors with magnitudes 3\sqrt{3}, 1, 2 respectively, such that a⃗×(a⃗×c⃗)+3b⃗=0⃗\vec{a}\times(\vec{a}\times\vec{c})+3\vec{b}=\vec{0}, if θ\theta is the angle between a⃗\vec{a} and c⃗\vec{c}, then sec⁡2θ\sec^2\theta is

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  4. Question 4

    13 Apr 2026 S1

    86
    Let aˉ,bˉ,cˉ\bar{a}, \bar{b}, \bar{c} be three vectors having magnitudes 1, 1 and 2 respectively. If aˉ×(aˉ×cˉ)+bˉ=0ˉ\bar{a} \times (\bar{a} \times \bar{c}) + \bar{b} = \bar{0}, then the acute angle between aˉ\bar{a} and cˉ\bar{c} is

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  5. Question 5

    18 Apr 2026 S2

    91
    If ∣aˉ∣=∣bˉ∣=1,∣cˉ∣=2|\bar{a}| = |\bar{b}| = 1, |\bar{c}| = 2 and aˉ×(aˉ×cˉ)+bˉ=0ˉ\bar{a} \times (\bar{a} \times \bar{c}) + \bar{b} = \bar{0}, then the acute angle between aˉ\bar{a} and cˉ\bar{c} is …\ldots

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