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a = i - j + 2k, b = 2i + 4j + k, c = mi + j + nk mutually perpendicular: find m and n

The same question, asked 3 times: 2 May 2023 S1, 11 May 2023 S2, 15 May 2023 S1. Every wording below, oldest first.

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  1. Question 1

    2 May 2023 S1

    114
    If the vectors a⃗=i^−j^+2k^\vec{a} = \hat{i} - \hat{j} + 2\hat{k}, b⃗=2i^+4j^+k^\vec{b} = 2\hat{i} + 4\hat{j} + \hat{k} and c⃗=mi^+j^+nk^\vec{c} = m\hat{i} + \hat{j} + n\hat{k} are mutually perpendicular, then (m,n)(m, n) is

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  2. Question 2

    11 May 2023 S2

    147
    If the vectors a⃗=i^−j^+2k^\vec{a}=\hat{i}-\hat{j}+2\hat{k}, b⃗=2i^+4j^+k^\vec{b}=2\hat{i}+4\hat{j}+\hat{k} and c⃗=λi^+j^+μk^\vec{c}=\lambda\hat{i}+\hat{j}+\mu\hat{k} are mutually orthogonal, then (λ,μ)≡(\lambda,\mu)\equiv

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  3. Question 3

    15 May 2023 S1

    145
    If the vectors a⃗=i^−j^+2k^\vec{a}=\hat{i}-\hat{j}+2\hat{k}, b⃗=2i^+4j^+k^\vec{b}=2\hat{i}+4\hat{j}+\hat{k} and c⃗=pi^+j^+qk^\vec{c}=p\hat{i}+\hat{j}+q\hat{k} are mutually orthogonal, then (p,q)(p,q) is equal to

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