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b x c = b x a and c.a = 0 for given a and b: value of c.b

The same question, asked 2 times: May 2022 S1*, 14 May 2024 S2. Every wording below, oldest first.

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  1. Question 1

    May 2022 S1

    116
    Let a⃗=i^−2j^+k^\vec{a} = \hat{i} - 2\hat{j} + \hat{k} and b⃗=i^−j^+k^\vec{b} = \hat{i} - \hat{j} + \hat{k} be two vectors. If c⃗\vec{c} is a vector such that b⃗×c⃗=b⃗×a⃗\vec{b} \times \vec{c} = \vec{b} \times \vec{a} and c⃗⋅a⃗=0\vec{c} \cdot \vec{a} = 0, then c⃗⋅b⃗\vec{c} \cdot \vec{b} is equal to

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  2. Question 2

    14 May 2024 S2

    120
    Let a⃗=i^+2j^−k^\vec{a} = \hat{i}+2\hat{j}-\hat{k} and b⃗=i^+j^−k^\vec{b} = \hat{i}+\hat{j}-\hat{k}. If c⃗\vec{c} satisfies b⃗×c⃗=b⃗×a⃗\vec{b}\times\vec{c} = \vec{b}\times\vec{a} and c⃗⋅a⃗=0\vec{c}\cdot\vec{a} = 0, then c⃗⋅b⃗\vec{c}\cdot\vec{b} is

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