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Vector in the plane of a and b whose projection on c is 1/sqrt(3)

The same question, asked 3 times: May 2022 S1*, 15 May 2023 S1*, 10 May 2024 S1. Every wording below, oldest first.

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  1. Question 1

    May 2022 S1

    123
    If a⃗=i^+j^+k^\vec{a} = \hat{i}+\hat{j}+\hat{k}, b⃗=i^−j^+k^\vec{b} = \hat{i}-\hat{j}+\hat{k} and c⃗=i^−j^−k^\vec{c} = \hat{i}-\hat{j}-\hat{k} are three vectors then vector r⃗\vec{r} in the plane of a⃗\vec{a} and b⃗\vec{b}, whose projection on c⃗\vec{c} is 13\frac{1}{\sqrt{3}}, is given by

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  2. Question 2

    15 May 2023 S1

    108
    Let a⃗=i^+j^+k^\vec{a}=\hat{i}+\hat{j}+\hat{k}, b⃗=i^−j^+k^\vec{b}=\hat{i}-\hat{j}+\hat{k} and c⃗=i^−j^−k^\vec{c}=\hat{i}-\hat{j}-\hat{k} be three vectors. A vector v⃗\vec{v} in the plane of a⃗\vec{a} and b⃗\vec{b}, whose projection on c⃗\vec{c} is 13\frac{1}{\sqrt{3}}, is given by

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  3. Question 3

    10 May 2024 S1

    135
    If a⃗=i^+2j^+k^\vec{a}=\hat{i}+2\hat{j}+\hat{k}, b⃗=i^−j^+k^\vec{b}=\hat{i}-\hat{j}+\hat{k} and c⃗=i^+j^−k^\vec{c}=\hat{i}+\hat{j}-\hat{k}, then a vector in the plane of a⃗\vec{a} and b⃗\vec{b}, whose projection on c⃗\vec{c} is 13\frac{1}{\sqrt{3}}, is

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