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[a b c] for a = i - k, b = xi + j + (1 - x)k, c = yi + xj + (1 + x - y)k: on which of x, y does it depend

The same question, asked 4 times: May 2022 S1, 10 May 2023 S1, 21 Apr 2025 S1, 18 Apr 2026 S1. Every wording below, oldest first.

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  1. Question 1

    May 2022 S1

    105
    If a⃗=i^−k^\vec{a} = \hat{i} - \hat{k}, b⃗=xi^+j^+(1−x)k^\vec{b} = x\hat{i} + \hat{j} + (1-x)\hat{k} and c⃗=yi^+xj^+(1+x−y)k^\vec{c} = y\hat{i} + x\hat{j} + (1+x-y)\hat{k}, then [a⃗ b⃗ c⃗][\vec{a}\ \vec{b}\ \vec{c}] depends on

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  2. Question 2

    10 May 2023 S1

    104
    If a⃗=i^−k^, b⃗=xi^+j^+(1−x)k^\vec{a}=\hat{i}-\hat{k},\,\vec{b}=x\hat{i}+\hat{j}+(1-x)\hat{k} and c⃗=yi^+xj^+(1+x−y)k^\vec{c}=y\hat{i}+x\hat{j}+(1+x-y)\hat{k}, then a⃗⋅(b⃗×c⃗)\vec{a}\cdot(\vec{b}\times\vec{c}) depends on

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  3. Question 3

    21 Apr 2025 S1

    118
    Three vectors i^−k^,λi^+j^+(1−λ)k^\widehat{i}-\widehat{k},\lambda\widehat{i}+\widehat{j}+ (1 -\lambda)\widehat{k} and μi^+λj^+(1+λ−μ)k^\mu\widehat{i}+\lambda\widehat{j}+ (1 +\lambda-\mu)\widehat{k} represents coterminous edges of a parallelopiped, then the volume of the parallelopiped depends on.

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  4. Question 4

    18 Apr 2026 S1

    88
    If aˉ=i^−k^,bˉ=xi^+j^+(1−x)k^\bar{a} = \hat{i} - \hat{k}, \bar{b} = x\hat{i} + \hat{j} + (1 - x)\hat{k} and cˉ=yi^+xj^+(1+x−y)k^\bar{c} = y\hat{i} + x\hat{j} + (1 + x - y)\hat{k} then [aˉ bˉ cˉ][\bar{a}\ \bar{b}\ \bar{c}] depends on

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