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Point of intersection of the lines r x a = b x a and r x b = a x b

The same question, asked 3 times: 11 May 2024 S1*, 20 Apr 2025 S2*, 13 Apr 2026 S2. Every wording below, oldest first.

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  1. Question 1

    11 May 2024 S1

    141
    If a⃗=i^+j^\vec{a}=\hat{i}+\hat{j}, b⃗=2j^−k^\vec{b}=2\hat{j}-\hat{k} and r⃗×a⃗=b⃗×a⃗\vec{r}\times\vec{a}=\vec{b}\times\vec{a}, r⃗×b⃗=a⃗×b⃗\vec{r}\times\vec{b}=\vec{a}\times\vec{b}, then the value r⃗∣r⃗∣\frac{\vec{r}}{|\vec{r}|} is

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  2. Question 2

    20 Apr 2025 S2

    104
    If aˉ=i^+j^,bˉ=2i^−k^\bar{a}=\widehat{i}+\widehat{j},\bar{b}= 2\widehat{i}-\widehat{k} then the point of intersection of the lines r‾×a‾=b‾×a‾\overline{r}\times\overline{a}=\overline{b}\times\overline{a} and r‾×b‾=a‾×b‾\overline{r}\times\overline{b}=\overline{a}\times\overline{b} is

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  3. Question 3

    13 Apr 2026 S2

    88
    If aˉ=i^+j^\bar{a} = \hat{i} + \hat{j} and bˉ=i^−k^\bar{b} = \hat{i} - \hat{k}, then the point of intersection of the lines rˉ×aˉ=bˉ×aˉ\bar{r} \times \bar{a} = \bar{b} \times \bar{a} and rˉ×bˉ=aˉ×bˉ\bar{r} \times \bar{b} = \bar{a} \times \bar{b} is

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