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Vector c with a . c = |c|, |c - a| given and a given angle with a x b: find |(a x b) x c|

The same question, asked 3 times: 2021 Paper 11*, 6 Apr 2024*, 28 Jan 2025. Every wording below, oldest first.

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  1. Question 1

    2021 Paper 11

    54
    Let a→=2i^+j^−2k^\overrightarrow{a}= 2\widehat{i}+\widehat{j}- 2\widehat{k} and b→=i^+j^\overrightarrow{b}=\widehat{i}+\widehat{j}. If c→\overrightarrow{c} is a vector such that a→⋅c→=∣c→∣,∣c→−a→∣=22\overrightarrow{a}\cdot\overrightarrow{c}= |\overrightarrow{c}|,|\overrightarrow{c}-\overrightarrow{a}| = 2\sqrt{2} and the angle between (a→×b→)(\overrightarrow{a}\times\overrightarrow{b}) and c→\overrightarrow{c} is π6\frac{\pi}{6}, then the value of ∣(a→×b→)×c→∣|(\overrightarrow{a}\times\overrightarrow{b}) \times\overrightarrow{c}| is:

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  2. Question 2

    6 Apr 2024

    161
    Let a→=6i^+j^−k^\overrightarrow{a}= 6\widehat{i}+\widehat{j}-\widehat{k} and b→=i^+j^\overrightarrow{b}=\widehat{i}+\widehat{j}. If c→\overrightarrow{c} is a vector such that ∣c→∣≥6,a→⋅c→=6∣c→∣,∣c→−a→∣=22|\overrightarrow{c}| \geq 6,\overrightarrow{a}\cdot\overrightarrow{c}= 6|\overrightarrow{c}|,|\overrightarrow{c}-\overrightarrow{a}| = 2\sqrt{2} and the angle between a→×b→\overrightarrow{a}\times\overrightarrow{b} and c→\overrightarrow{c} is 60∘60^{\circ}, then ∣(a→×b→)×c→∣|(\overrightarrow{a}\times\overrightarrow{b}) \times\overrightarrow{c}| is equal to:

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  3. Question 3

    28 Jan 2025

    73
    Let a→=i^+j^+k^,b→=2i^+2j^+k^\overrightarrow{a}=\widehat{i}+\widehat{j}+\widehat{k},\overrightarrow{b}= 2\widehat{i}+ 2\widehat{j}+\widehat{k} and d→=a→×b→\overrightarrow{d}=\overrightarrow{a}\times\overrightarrow{b}. If c→\overrightarrow{c} is a vector such that a→⋅c→=∣c→∣,∣c→−2a→∣2=8\overrightarrow{a}\cdot\overrightarrow{c}= |\overrightarrow{c}|,|\overrightarrow{c}- 2\overrightarrow{a}|^{2}= 8 and the angle between d→\overrightarrow{d} and c→\overrightarrow{c} is π4\frac{\pi}{4}, then ∣10−3 b→⋅c→∣+∣d→×c→∣2|10 - 3\overrightarrow{\text{ }b}\cdot\overrightarrow{c}| + |\overrightarrow{d}\times\overrightarrow{c}|^{2} is equal to ….\ldots..