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c is coplanar with a and b, perpendicular to b, and a . c is given: find c

The same question, asked 3 times: 2021 Paper 1*, 28 Jun 2022*, 24 Jan 2025. Every wording below, oldest first.

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  1. Question 1

    2021 Paper 1

    87
    Let three vectors a→,b→\overrightarrow{a},\overrightarrow{b} and c→\overrightarrow{c} be such that c→\overrightarrow{c} is coplanar with a→\overrightarrow{a} and b→,a→.c→=7\overrightarrow{b},\overrightarrow{a}.\overrightarrow{c}= 7 and b→\overrightarrow{b} is perpendicular to c→\overrightarrow{c}, where a→=−i^+j^+k^\overrightarrow{a}= -\widehat{i}+\widehat{j}+\widehat{k} and b→=2i^+k^\overrightarrow{b}= 2\widehat{i}+\widehat{k}, then the value of 2∣a→+b→+c→∣22\left|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right|^{2} is

  2. Question 2

    28 Jun 2022

    84
    If a→=2i^+j^+3k^, b→=3i^+3j^+k^\overrightarrow{a}= 2\widehat{i}+\widehat{j}+ 3\widehat{k},\ \overrightarrow{b}= 3\widehat{i}+ 3\widehat{j}+\widehat{k} and c→=c1i^+c2j^+c3k^\overrightarrow{c}=c_{1}\widehat{i}+c_{2}\widehat{j}+c_{3}\widehat{k} are coplanar vectors and a→⋅c→=5,b→⊥c→\overrightarrow{a}\cdot\overrightarrow{c}= 5,\overrightarrow{b}\bot\overrightarrow{c}, then 122(c1+c2+c3)122\left( c_{1}+c_{2}+c_{3} \right) is equal to

  3. Question 3

    24 Jan 2025

    51
    Let a→=i^+2j^+3k^,b→=3i^+j^−k^\overrightarrow{a}=\widehat{i}+ 2\widehat{j}+ 3\widehat{k},\overrightarrow{b}= 3\widehat{i}+\widehat{j}-\widehat{k} and c→\overrightarrow{c} be three vectors such that c→\overrightarrow{c} is coplanar with a→\overrightarrow{a} and b→\overrightarrow{b}. If the vector c→\overrightarrow{c} is perpendicular to b→\overrightarrow{b} and a→.c→=5\overrightarrow{a}.\overrightarrow{c}= 5, then ∣c→∣|\overrightarrow{c}| is equal to

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