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a = i + j + k, a x b = j - k and a.b given: find b

The same question, asked 3 times: 9 May 2023 S1*, 9 May 2024 S2, 25 Apr 2025 S1. Every wording below, oldest first.

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  1. Question 1

    9 May 2023 S1

    137
    If a⃗=i^+j^+k^\vec{a}=\hat{i}+\hat{j}+\hat{k}, a⃗⋅b⃗=1\vec{a}\cdot\vec{b}=1 and a⃗×b⃗=j^−k^\vec{a}\times\vec{b}=\hat{j}-\hat{k}, then b⃗\vec{b} is

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  2. Question 2

    9 May 2024 S2

    119
    a⃗=i^+j^+k^,  b⃗=j^−k^\vec{a} = \hat{i}+\hat{j}+\hat{k},\;\vec{b} = \hat{j}-\hat{k}, then vector r⃗\vec{r} satisfying a⃗×r⃗=b⃗\vec{a}\times\vec{r} = \vec{b} and a⃗⋅r⃗=3\vec{a}\cdot\vec{r} = 3 is

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  3. Question 3

    25 Apr 2025 S1

    105
    If a‾=i^+j^+k^,b‾=j^−k^\overline{a}=\widehat{i}+\widehat{j}+\widehat{k},\overline{b}=\widehat{j}-\widehat{k} then a vector c‾\overline{c} such that a‾×c‾=b‾\overline{a}\times\overline{c}=\overline{b} and a‾⋅c‾=3\overline{a}\cdot\overline{c}= 3 is

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