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a = 2i + j - 2k, b = i + j, |c - a| and |(a x b) x c| given with the angle between c and a x b: find a.c

The same question, asked 3 times: 2 May 2023 S1*, 4 May 2023 S2*, 11 May 2024 S2. Every wording below, oldest first.

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  1. Question 1

    2 May 2023 S1

    104
    Let a⃗=2i^+j^−2k^\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k} and b⃗=i^+j^\vec{b} = \hat{i} + \hat{j}. Let c⃗\vec{c} be a vector such that ∣c⃗−a⃗∣=3|\vec{c} - \vec{a}| = 3 and ∣(a⃗×b⃗)×c⃗∣=3|(\vec{a} \times \vec{b}) \times \vec{c}| = 3 and the angle between c⃗\vec{c} and a⃗×b⃗\vec{a} \times \vec{b} is 30∘30^\circ, then a⃗⋅c⃗\vec{a} \cdot \vec{c} is equal to

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  2. Question 2

    4 May 2023 S2

    122
    Let a⃗=2i^+j^−2k^\vec{a}=2\hat{i}+\hat{j}-2\hat{k} and b⃗=i^+j^\vec{b}=\hat{i}+\hat{j}. Let c⃗\vec{c} be a vector such that ∣c⃗−a⃗∣=3|\vec{c}-\vec{a}|=3 and ∣(a⃗×b⃗)×c⃗∣=3|(\vec{a}\times\vec{b})\times\vec{c}|=3 and the angle between c⃗\vec{c} and a⃗×b⃗\vec{a}\times\vec{b} is 30°, then a⃗⋅c⃗\vec{a}\cdot\vec{c} is equal to

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  3. Question 3

    11 May 2024 S2

    139
    Let a⃗=2i^+j^−2k^\vec{a}=2\hat{i}+\hat{j}-2\hat{k}, b⃗=i^+j^\vec{b}=\hat{i}+\hat{j} and c⃗\vec{c} be a vector such that ∣c⃗−a⃗∣=4|\vec{c}-\vec{a}|=4, ∣(a⃗×b⃗)×c⃗∣=3|(\vec{a}\times\vec{b})\times\vec{c}|=3 and the angle between c⃗\vec{c} and a⃗×b⃗\vec{a}\times\vec{b} is π6\frac{\pi}{6}, then a⃗⋅c⃗\vec{a}\cdot\vec{c} is equal to

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